A freewheel packages the ratcheting mechanism and the cogs into one component that screws onto the hub over the end of the axle. A cassette is just the cogs, which slide over the cassette body and are held in place by a threaded lock ring (or sometimes the last cog), which provides the ratcheting mechanism and is an integral part of the hub.
The definitive way to tell what you have is to remove the rear wheel and look at it closely. Freewheels have two main components, the core that screws onto the hub and the shell that spins with the cogs. When you spin the cogs on a freewheel, you can see the shell spinning around the stationary core. On a hub that uses a cassette, also called a freehub, the core is inside the hub where you can't see it.
Cassettes (freehubs) became prevalent for better bikes because they provide the closer tolerances needed for indexed shifting and more cogs, and because they provide for a stronger hub.
These days most bikes use 8-, 9-, and 10-speed freehubs, with 5-speed freewheels on cheap discount store bikes, 6-speed freewheels on kids' bikes, and 7-speed freewheels on entry level adult bikes. Most shops don't stock freewheels for every application, but they are available. 7-speed freehubs are a rarity because it simply didn't take that long to modify the design to accommodate 8 and 9 speeds. Cassettes for some but not all 7-speed freehubs are still available.
If you don't understand, take the wheel to a shop and they will tell you what you have.
