Free Play - Forza Motorsport 4
Free Play Mode in Forza 4 includes the menu selections for Quick Race, Split Screen, and Hot Lap.
Quick Race is just what it sounds like -- a mode in which you can pick any car that is in the game including special cars, and then any track you have access to, and race. Note that unlike online and career races you do not earn either XP or Credits from the races in this mode, but you can unlock Achievements here.
Since you have access to virtually every car in the game in this mode, this is the perfect place for you to go and evaluate cars before you buy them, get to know them, and decide if you want them. Obviously you are going to want every car in the game eventually, but as you work your way through Career Mode knowing which cars you like more than others can help in choosing your rides!
Split Screen is what it sounds like -- a local mode for two-player racing with the screen split in two. To use this mode you need a second controller plugged into the game, and another human player logged in.
Training with the Hot Lap Mode
Hot Lap -- In this mode you can choose any car, any track, and then run laps, with the idea being to practice the tracks and improve your lap time. How this works (as shown in the video) is really simple: the first lap you run sets the time, and then when you begin the next lap a "ghost" of your previous run spawns, and you can then see how much of an improvement or how worse off your next lap is.
Using this you can feel out the right path to take on a given track, which makes it very useful for training specific tracks when you are looking for consistent and good results.