There are five main categories of bits. These are:
Snaffle
Curb
Pelham
Gag
Nose
In dressage, there are restrictions on the type of bit you can use. Only snaffle bits, and in higher levels, double bridles, may be used and are classed as 'legal'.
Of course there are literally thousands of different kinds of bits, enough to fill many books. But the basics are the snaffle (broken) and solid or curb (unbroken). It's not as simple as western bits...