A BMX bike is the shorthand version of bicycle motocross bike. Why they feel the need to say bike twice I don’t know. BMX started slightly later than mountain biking, but under similar circumstances. Mountain biking attempted to build bikes that could handle rough terrain with aplomb; BMX sought to build a bike sturdy enough to perform jumps and tricks. This distinction led to a rather different evolutionary path for each style of cycling, though they have borrowed from each other along that path.
The first BMX bikes were built to race down the fire roads of the mountains in California. They didn’t come with shocks, and in fact shock absorbers have been eschewed by the vast majority of the sub-genre of BMX. The reason given is control and stability. What does that mean, you ask?
Well since inception, BMX bikes have become used as the primary vehicle for bike tricks. The tricks involve acts of great balance and coordination, and the stability of your BMX bike becomes a big factor in the successful completion of a trick. When you are balancing the weight of your bike and your body on your front fork, you don’t want that fork moving… well that’s exactly what a suspension is designed to do; so many bikers have simply taken them out.