A rig consists of a container, harness, one or two canopies, and possibly an AAD.
"Containers" are the backpack-like things that your "canopies" (parachutes) are kept in when you jump out of a plane.
"Harnesses" are the straps that keep the container (and canopy) attached to you. Most containers are permanently attached to a harness.
A "canopy" is the technical term for the actual parachute. The "container" is not part of the canopy, and is often purchased separately.
Most containers intended for sports parachuting hold two canopies, a main and a reserve. This means that if there is a problem with the main canopy, you can cut it away and open the reserve.
An "AAD" or "Automatic Activation Device" is a safety device to automatically open your reserve parachute at the last moment if you do not open your parachute normally.
(Note that "Skydiving" and "Parachuting" mean the same thing).