The equipment used to ride a horse is called tack.  Here in the US, equestrian sports can be divided into two broad categories, Western, and English.  The Western activities are mainly modeled after those traits desired in horses used to drive cattle.  A short quick horse which can head off stray calves is a must.  The breed most desired for the western sports is the American Quarter Horse.  Below is an example of a western saddle which would be used on such a horse:

On the other side of the pond (the Atlantic) Europe refined it's equestrian sports in what is known today as the English style of riding.  Sports such as Hunter/Jumper, Dressage, and Eventing arose out of sports such as fox hunting.  Below is a picture of a typical English Saddle:

Bridles are essentially the same in both disciplines, although the bit (the part that goes in the horse's mouth) varies dramatically.  Here is a picture of a common bridle: