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. This sequence is called the transposition, docking, and extraction maneuver.
The S-IVB and spacecraft then go their separate ways and the docked CSM/LM coasts to the moon, over a flight of about 3 days. During the flight to and from the moon, the spacecraft would be rotating about its long axis to maintain even heating from sunlight (this is the PTC, "passive thermal control" mode, sometimes known as the "barbecue roll.").
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.
The LM stays on the lunar surface for between one and three days, with the commander and LMP generally doing one EVA per day, while the CSM in orbit overhead circles the moon every 2 hours. At the end of the stay, they discard unnecessary equipment (like the big life-support backpacks). 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, and 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 to slow down for a safe splash-down in the ocean.
This is of course a rather loose overview of a very complex process! Several apollo-program Q/As here go into various aspects in more detail, so you might enjoy perusing them.