The pin bridge. Bridge pins wedge the ball end of the string firmly against the guitars bridge and bridge plate. When restringing an acoustic guitar or bass with bridge pins it's important that the ball end of the string sits firmly against the bridge plate.
When installing strings on a pin bridge place the ball end of the string in the bridge pin hole, place the pin in the hole and pull up on the string until the ball end sits firmly against the bridge plate. This will prevent the string from slipping as the tension is raised which causes constant retuning until the string is seated. More about bridge pins.

The pinless (or top loading) bridge. The strings simply pass thru the back side of this bridge and do not require pins. When stringing this bridge you will find that it is easier if you put a "hook like" bend on the tip of the string before threading it thru the hole, this way it feeds up and over the saddle.

The tailpiece. Strings hook or pass thru holes in these tailpieces which are commonly referred to as "trapeze tailpieces"...think circus.

The Tie block / Tie Back or classical guitar style bridge - These bridges are most often seen on classical guitars. Instead of ball ends on the ends of the string, the string itself is identical on each end and is tied to the bridge.