If there was ever a thread tailor-made for an Aerospace Engineer racing fanatic! Anyway, when two cars are drafting, the front car blocks the free airstream from impinging on the front of the drafting vehicle, so the drag is less. Also, the trailing car fills in the area behind the lead car, which reduces the vortex behind the lead car and the drag is less for the leader as well, but not as much as for the trailing car.
Drafting also effects downforce and handling. Sometimes you’ll hear drivers comment about how their car handles a certain way in the draft, but a different way by itself. Since the air is no longer pushing on the front of the trailing car, front downforce is reduced and the car will start to understeer (or “push”). That is, the car will not want to turn as well. For the lead car, the effect is to lift the airstream off of the spoiler and cause the car to oversteer (also called “loose”). The back of the car will want to snap around and spin the car in a turn. (Drivers sometimes take advantage of this by getting the lead car a little loose, forcing the driver to check up a bit.)
The net result is that two cars running together on a high-speed oval (such as Daytona, for example) will lap faster than a single car running alone. If there is a group of cars drafting in line and a car steps out of the draft, it will generally fall back until it can get back in line. Interestingly, three cars will be a little bit faster than two, and four a bit faster than three, beyond which you get diminishing returns (the reasons are a bit complicated for this post, but it has to do with the ratio of overall length to cross sectional area getting longer). This is why you’ll often see a lead pack of three or four cars breaking away from the rest of the group.
Sometimes a car can also use the draft to slingshot around another car. Basically, it is taking advantage of the reduced drag to run up on the lead car and build up enough momentum to pull out and pass the lead car before the increased drag of running in the full airstream slows it back down to it’s normal speed. This requires careful timing to pull off successfully.
Then there is bump drafting, where the trailing car will actually run up and bump the lead car. This will give the leader an additional 1-2 mph boost and pull the trailing car along with it. (The danger is all this bumping might close up your air inlets and cause the car to overheat).
All of this assumes you’re talking about NASCAR-style cars. For high-downforce cars, such as a Formula 1 or Indy car, the turbulence and reduced downforce upset the trailing car so much, they generally don’t form the drafting packs you see in stock car racing. Usually they use drafting (which is generally called slipstreaming in European racing) just for slingshot passes.
Also, you don’t have to be at race-car speeds to experience drafting. Bicycles can draft as you’ve noted (which is why these races tend to form a large peloton). I race in triathlons and during the swim (which is a mass start) you can even draft off of other swimmers.
(On preview, I notice CJJ* beat me to it, but since my post has some information his doesn’t, I’ll post anyway.)