For most people, Saxony carpet is the standard carpet that comes to mind when they think of carpet. It is your standard cut pile carpet that is found in many homes.
Cut and loop is a carpet style that was incredibly popular in the 70s and 80s that have seen some resurgence in recent years. It combines both cut pile carpet with loop pile in interesting geometric and abstract patterns.
Triexta carpet is a relatively new carpet fiber that is incredibly, soft, durable, and stain resistant. Here are the pros and cons.