The Apollo spacecraft consists of three major parts:
The Command Module (CM), a conical module where the three crew members live during launch from Earth and travel to and from the moon, and which re-enters Earth's atmosphere alone at the end of the trip;
The Service Module (SM), a cylindrical section containing fuel, power, life support, communications, a big rocket engine, and other components;
The Lunar Module (LM), a "spidery-looking" craft which lands on the moon carrying two of the crew members. The LM in turn is made up of a lower section called the descent stage and an upper section, the ascent stage. Both stages land on the moon, and only the ascent stage returns.
The Command and Service Modules remain mated for all but the very last minutes of the mission, so the combination is referred to as the Command/Service Module or CSM.
During launch (which I won't detail) the LM is tucked behind the CSM under a conical fairing attached to the Saturn V 3rd stage (the S-IVB). The S-IVB is the part that propels the spacecraft out of low Earth orbit and on its way to the moon.
The fairing opens up and the CSM detaches, turns around, and docks to the LM. The front of the CM has a hatch that mates with a hatch on the top of the LM. After docking, the CSM pulls the LM away from the S-IVB, and launcher and spacecraft go their separate ways. This sequence is called the transposition, docking, and extraction maneuver.
Once the spacecraft reaches the moon, the large engine on the Service Module is used to enter a circular orbit around the moon. The commander and LM pilot enter the LM, undock, and use the engine on the descent stage to reach the surface of the moon. The CM pilot remains in orbit around the moon.
To return, the LM ascent stage lifts off, using the descent stage as a launch pad, and arcs into orbit around the moon.
Once in lunar orbit, the ascent stage meets up with the CSM (discussed a bit here), and docks again, and the crew return to the Command Module.
The LM ascent stage is discarded, the CSM fires the big engine again to return to Earth.
The back side of the Command Module sports a thick heat shield; detaching from the Service Module, the CM turns blunt-end-first and hits Earth's atmosphere, using the resistance of the atmosphere to slow down, and finally deploys parachutes for a safe landing in the ocean.
This is of course a rather loose overview of a very complex process! Several Q/As here with the apollo-program tag go into various aspects in more detail, so you might enjoy perusing them.