Downhill boots are any boot that works with standard ski bindings. The only boots Skis.com sells that are not considered downhill boots are for Alpine Touring.
Side Country is a popular term that defines ski boots that may be used for walking or hiking up the mountain for access slopes with out chairlifts. Side Country ski boots typically have a ski/walk/hike switch that can make it easier for you to get around and many have a gripped sole for the more adventurous skier that will spend time seeking out fresh snow off the beaten path.
Freestyle boots are more upright in the cuff and stance to make skiing and landing switch easier. They also have a more relaxed fit with shock absorbing properties for sticking huge landings off of sick jumps.
Race boots are designed to be going fast at all times. They are focused on speed and performance with an aggressive stance, and thicker shells, with dense liners to deliver power and energy transfer.
Alpine Touring boots are designed to be lightweight and easy to hike, climb and skin in. They go up as easy as they come down, and are only compatible with AT bindings.