The ISS orbits around the world. Do astronauts fly the ISS on Earth's orbit like a plane pilot? If it wanted, could the ISS go to a specific part of the world apart from the the usual orbit? Does the ISS need powerful fuel while orbiting?
Most of the time, no one is piloting the ISS.
In general, an object in orbit stays in the same orbit without needing to be propelled or piloted.
At the relatively low altitude of the ISS's orbit, there's a small amount of atmospheric drag, slowing the station down and lowering its orbit, so about once a month they do a "reboost" maneuver to raise it back up. This does require a fair amount of fuel. In addition, the ISS occasionally has to make smaller maneuvers to avoid space debris.
The reboosts are usually performed by an uncrewed Progress spacecraft docked to the ISS. These reboost maneuvers are normally executed by computer under control from the ground, not directly by an astronaut aboard the ISS. The avoidance maneuvers are normally executed with thrusters on the Russian Zvezda module; these can be ground-controlled or locally controlled.
The Earth is rotating beneath the fixed orbit of the ISS. The orbit of the ISS is inclined at 51° relative to the equator, so at some point in time, the station will pass over any point on Earth between 51° north and 51° south latitude, so there's no need to maneuver there.
Changing the inclination of the orbit would take a very large amount of fuel. The current inclination is chosen to allow Russia to launch service missions to the station without overflying China. Increasing the inclination to reach higher latitudes would reduce the available payload in service missions; lower inclinations take advantage of the Earth's eastward rotation.
The ISS doesn't really change its orbital orientation. The angle of its orbit (51.6 degrees) was specifically chosen to make it relatively accessible to both US launches out of Kennedy and Russian launches from Baikonur, so they don't maneuver it around to fly over different parts of the earth. That said, because that orbit has a fairly high angle, the ISS will eventually pass over every part of the planet between 51.6 N and 51.6 S. That covers most of the occupied parts of the planet outside of Russia, Scandinavia, and upper Canada (and anywhere else equally far north).
Occasionally small amounts of thrust are applied to lift the station out of the way of debris, and the ISS needs to re-boost its orbit regularly, because at that altitude it loses about 100 meters per day due to atmospheric drag from the extremely diffuse air molecules. It has engines of its own in the Zvezda service module, or it can maneuver by having a docked vessel fire its engines.
NASA's 2014 publication about the ISS contains a lot of information about guidance and control. Chapter 8 on Debris Avoidance Maneuvers (DAM) is particularly notable. The section on p.148-149 of that publication explains a lot about ISS maneuvering operations, and I'll paraphrase to directly address your question.
When a maneuver is needed (whether DAM or a standard reboost), it's usually programmed by the Mission Control Center in Moscow (Russia controls the Zvezda service module, which has the engines in it), and the instruction packet is sent directly to Zvezda's computer (or the cargo pod's) from a ground station transmitter; no action is required from the crew to make the station do its thing, though they'll of course be informed to expect rotation or thrust. The crew can manually trigger pre-planned maneuvers if necessary*, but there's no input as direct as a joystick-and-throttle arrangement. Nobody flies it as such, it's just a series of computer instructions that say "at thus-and-such time, point in this direction and fire the engine at this throttle setting for this many seconds".
*There is supposedly no way for the astronauts to write custom control routines, but they have a few pre-written emergency maneuver packages ("PDAMs" - Predetermined Debris Avoidance Maneuvers) that they can trigger without ground control's input. The lack of manual control is probably because maneuvers have to be carefully laid out to avoid causing damage to the station by applying forces in just the wrong way. In case of a really catastrophic emergency, the plan would be to abandon the station rather than try to hand-fly it out of danger.
Like any spacecraft ISS does need to be navigated or “flown”. If nothing was done the ISS would not hold the attitude position that it needs to successfully operate and it would likely start tumbling. As stated by an Airbus spokesperson in an interview about the company's Detumbler device:
Dead satellites, especially in low Earth orbit (LEO), often end up tumbling, which is natural behavior due to orbital flight dynamics.
Due to the massive size of ISS it’s possible that some natural gravity gradient stabilization would occur, which would keep one side of ISS pointing downwards and eliminate or at least reduce tumbling. But even if that did happen it would not necessarily be in an ideal orientation.
For this reason the most commonly used type of ISS maneuvering is attitude control, the same concept as an airplane pilot keeping the wings level. However whereas an airplane generally needs to be kept pointed in the same direction that it is flying (and preferably right side up), a spacecraft can face any way it wants to with little or no effect on its path. ISS orientation is adjusted as needed for various optimizations including providing light to the solar panels, or being in the correct position for arriving or departing spacecraft.
However unlike space capsules or the Space Shuttle, ISS is not flown by astronauts but by teams on the ground, similar to a satellite. Commands are sent by ground stations which are stored in the ISS computers and executed at the scheduled time. As explained in a 2001 CBS news article (reprinted on Spaceflight Now) from early in the ISS program:
Computers in the newly installed $1.4 billion Destiny laboratory module began controlling the International Space Station's orientation for the first time today, spinning up four massive, fuel-saving gyroscopes in a critical milestone for the orbiting complex. Assuming tests and checkout operations go well - and so far, the computer-driven gyros are performing flawlessly - day-to-day operational control of the station will shift from Russian flight controllers to NASA shortly after Atlantis departs Sunday.
Up until this point, the station's orientation has been controlled by a computer system in the Russian Zvezda command module that orders periodic rocket firings to nudge the 105-ton station into different attitudes as required.
By switching to gyro control, station crews can conserve propellant and avoid jarring rocket firings that would disturb sensitive microgravity experiments.
This computer automated control is similar also to how the SpaceX Crew Dragon capsule is maneuvered, however the big difference is that Dragon always has two astronauts on board in a pilot role. When Dragon is maneuvering the two pilot astronauts are closely monitoring, ready to take over manual control if the automated systems fail, although so far this has never happened with Dragon. As explained in a TechCrunch.com article about the first crewed Dragon flight in 2020:
NASA astronauts Doug Hurley took over manual control of the SpaceX Crew Dragon spacecraft on Saturday, shortly after the vehicle’s historic first launch from Cape Canaveral in Florida. Crew Dragon is designed to fly entirely autonomous throughout the full duration of its missions, including automated docking, de-orbit and landing procedures, but it has manual control systems in case anything should go wrong and the astronauts have to take over.
The Russian Soyuz and Chinese Shenzhou spacecraft are operated with a similar strategy of computer automated being the primary maneuvering method, with manual control by astronauts as a backup. As explained in a NASA Automated Rendezvous and Capture Review Executive Summary:
From discussions with Soviet engineers, it seems the docking process can be controlled either from the ground or from the active (docking) spacecraft's onboard computer. The unmanned Progress resupply ships regularly dock with the current MIR Space Station. The Soyuz T spacecraft incorporated the IGLA system, and the later Soyuz TM and Progress M Series spacecraft incorporated the KURS.
Cosmonaut pilots can take manual control if needed, they can even manually fly reentry using a special hand controller:
Soyuz spacecraft manual hand controller (Steve Jurvetson, via Wikimedia Commons, CC-BY-2.0)
In the text for the above photo the author states:
This controller is used by cosmonauts for manual return on Soyuz capsule. They can hold it close to their chest while taking many g's. Ed Lu (who flew Soyuz) told me that if you are using it, something has gone wrong with all of the control systems, and you have to guess as to the thrust vectors.
Under ideal end-of-mission situations, an automatic re-entry system will return the Soyuz vehicle and crew from space safely back to the ground. However, for certain hardware and software malfunctions, the crew will be required to manually fly the Soyuz back to Earth through the atmosphere. To do this, cosmonauts use a hand controller that varies the aerodynamic lift on the capsule. Their objective is to manipulate the lift forces on the Soyuz descent module such that they will land as close as possible to the designated site where the recovery team will be waiting for them.
The Chinese Shenzhou spacecraft also has both automatic and manual control, as explained in this Spaceflight Now article from 2012:
Three Chinese astronauts will temporarily depart their quarters inside the orbiting Tiangong 1 space lab early Sunday, backing away inside a Shenzhou spacecraft before pilot Liu Wang takes control of the capsule to complete the first manual docking in China's burgeoning space program.
The Shenzhou 9 astronauts rode their spacecraft to an automated docking with the Tiangong space lab Monday, two days after lifting off from northwest China atop a Long March rocket.
However ISS does not have a similar pilot role. For example in the list of the twelve crew members of the current Expedition 70 on ISS, all of the crew members are listed as Flight Engineer, other than the two astronauts who have taken turns being the expedition Commander.
Although in an extreme malfunction type of situation astronauts on board could in theory potentially have some involvement in controlling ISS, as mentioned in this answer.
The navigation capabilities of ISS are limited, most of the time it just plows along in the same orbital path. Generally the only time that it changes course is for collision avoidance if it is calculated that a piece of space debris will be coming too close to ISS, so as a precaution it slightly changes its path. Sort of like when you are driving and you move slightly to the side of your lane when passing a cyclist. These potentially hazardous situations are calculated by the U. S. Air Force Space Surveillance Network (SSN). According to an article about space debris on the Aerospace Corporation website:
The SSN has radar and optical sensors at various sites around the world … The sensors can determine which orbit the objects are in and that information is used to predict close approaches, reentries, and the probability of a collision. Other nations also run space object tracking systems.
There is actually a relatively large navigational maneuver which routinely occurs when ISS is reboosted to a higher altitude. This is because even at the roughly 250 mile (400 km) altitude where ISS orbits there are still some stray air molecules which it runs into constantly which slows it down, requiring a reboost every once in a while. However ISS does not have enough propulsion to do this maneuver on its own and requires visiting spacecraft to provide this propulsion. Previously the Space Shuttle did reboosts, currently reboosts are now only being done by visiting Russian Progress cargo capsules, although there are plans to start having other cargo craft do it also. In 2022 a test reboost was done by a Cygnus cargo spacecraft, as mentioned in this Northrup Grumman press release:
Northrop Grumman Corporation’s Cygnus cargo spacecraft successfully boosted the orbit of the International Space Station (ISS). Docked to the ISS since February, Cygnus fired its main onboard engine to adjust the orbit of the station to the desired altitude to support upcoming operations. The station orbits approximately 250 miles above earth and requires a periodic reboost.
This redundancy provided by Cygnus and possibly other spacecraft in the future is important because until now all major maneuvering of ISS has been done by the Russian segment of ISS, with the exception of the much smaller attitude adjustments which can be done using gyroscopes on the U.S. segment. As explained on this (archived) NASA web page:
All International Space Station propulsion is provided by the Russian Segment and Russian cargo spacecraft. Propulsion is used for station reboost, attitude control, debris avoidance maneuvers and eventual deorbit operations are handled by the Russian Segment and Progress cargo craft. The U.S. gyroscopes provide day-to-day attitude control or controlling the orientation of the station. Russian thrusters are used for attitude control during dynamic events like spacecraft dockings and provide attitude control recovery when the gyroscopes reach their control limits.
The NASA web page goes on to point out that even the Cygnus reboost requires the assistance of the Russian segment:
Northrop Grumman’s Cygnus is the only U.S. commercial spacecraft currently in testing to provide limited capability for future reboosts. This capability relies on the Russian Segment for attitude control during the small reboost. It does not currently have the capability to replace attitude control functions for the space station or carry adequate propellant for long-term sustained operations.
As for “can the ISS go to a specific part of the world apart from the the usual orbit?” it doesn’t really need to, as also mentioned in the answer its 51 degree orbital inclination takes it over nearly every part of the world on a daily basis.
There are already a few answers with really good information on how the ISS orbits, but your question asked "WHO" navigates/flies the ISS.
The answer is several flight controller positions working in coordination from various control centers on the ground.
The ADCO console (Attitude Determination and Control Officer) is the NASA position that manages the station orientation, along with their counterparts at ROSCOSMOS. They have a backroom position HawkI (pronounced Hawkeye) that helps with the heavy lifting for the detailed inertial calculations, while ADCO handles the coordination and execution of those calculations.
The TOPO console (Trajectory OPerations Officer) is the NASA position responsible for the trajectory of the ISS. They also coordinate with NORAD/Space Command to get information about possible conjunctions between the ISS and tracked space debris. For a high enough probability conjunction, a PDAM will be executed - usually commanded from the ground, not by a crew member.
Most of the time, Station is flying in +XVV/+ZLV TEA (Torque Equilibrium Attitude ) (PDF link). This is with NOD2 at the fore and the SM aft. Basically the torques imparted during each orbit (gravitational gradient and atmospheric drag) are cancelled out and result in a near net zero change in attitude.
For docking/berthing maneuvers with visiting vehicles, the ISS will often have to fly backwards or on its nose. These changes are reflected in the ATL (Attitude Timeline) maintained by ADCO. Many payloads (science experiments) sensitive to microgravity disturbances need safing prior to these maneuvers. The maneuvers are completed by OPM (Optimal Propellant Maneuver) which uses minimal thruster firings along with the CMGs (Control Moment Gyro) to slowly cartwheel Station to the desired attitude.
Wikipedia summary of ISS Flight Control Positions: https://en.wikipedia.org/wiki/Flight_controller#ISS_flight_control_positions_to_2010