Among all the other things the name has been used for, Fox is the name of a company which makes forks and shocks for mountain bikes, dirt bikes, snowmobiles and side x sides (a kind of off-road two-seat go-kart.) Fox is generally considered to make some of the best equipment especially in mountain biking, with a range of products with a variety of features.
The company was started by Bob Fox in 1974 to produce motocross motorcycle parts. Fox has long been a major name in motorcycle racing, especially starting with the first win on Fox Shocks in 1976. At the time, the old short-travel shocks were giving way to long-travel and most of the designs were based on the short-travel stuff; Fox made an air shock designed for long travel and the rest is history.
Today the major differentiators between Fox and others (at least in mountain biking) are the durability and ride quality. High-end Fox air forks feature an option called "TALAS" which provide separate high and low speed compression adjustments (rebound is separate) and their good rear shocks have an option called "ProPedal" which allows short, high-speed compression for floating bumps, but which prevents long, slow compression strokes to permit uphill pedaling on full suspension bicycles.