Basically, with an unanchored putter, you have six degrees of freedom; your fingers, wrists, elbows, shoulders, hips and knees can all move during a putt (pretty much any golf stroke) possibly resulting in a putt off the intended line, or too hard/soft.
A belly putter, designed to be anchored pretty much at the bellybutton, almost completely takes the fingers, wrists and shoulders out of it, and reduces the impact of the elbows and hips. A long putter (anchored to the chest or even the chin) also takes the fingers and hands out of it, and depending on its length and where it's anchored reduces the impact of movements in the hips and knees, primarily by allowing the golfer to stand more upright. The long putter, anchored from the chin, also gives the golfer a sight picture right down the shaft of the club which can help in centering the clubhead on the ball.
All of these putters can be used for the two most common types of putt; the pendulum swing and the power putt. In virtually all cases, the fingers, wrists and elbows should not move during the shot, and the shoulders should only move as much as they have to; what changes between the two putting styles, regardless of putter, is whether the club is "pushed" through the stroke or simply raised back and then allowed to let fall.