Hi Kyle!
Renting varies from place to place, but this is generally how it works. There is a schedule, either online or in the FBO, where a list of reservations are maintained. You pick out the date and times you need the plane, and your name appears on the schedule for that date and time. Of course, that depends on the aircraft's availability. Sometimes the aircraft has to go in for maintenance, it's been reserved for someone else, etc.
You are charged a rate that is usually based off of the Hobbs time. In most piston airplanes, flying hours are determined two ways: The tachometer time, or the Hobbs time. The Hobbs meter starts ticking when you turn on the aircraft battery, and the Tach time starts ticking when the engine RPM is revved up to a certain point. Most places charge by Hobbs, but you can be charged for tach at others. It just depends.
If you want to take an airplane out for an extended period of time, you will either encounter some sort of surcharge, or you will be charged the equivalent of a certain number of hours. For instance, out of my FBO, if I want to take a plane out for a 12-hour period, I get charged for a minimum of 3 hours of flight time, plus any additional flying time I do above 3 hours. If I take out a plane for 12 hours to do a two-hour flight, the FBO charges me for 3 hours of flight time, so it is economical for me to plan to fly for at least three hours.
Fuel is either included in the hourly rental cost; this is called renting "wet." When you rent "dry," you have to pay for the fuel, but the rental rate is usually cheaper.
Rental rates are dependent on the aircraft category, type, avionics, etc.