A golfer who hits the ball very gently close to the hole is a putter. The club used to do this is also a putter. But to putter is to poke around aimlessly, maybe watch a little golf on TV.
In golf, the putter is the club with the flat face, and sometimes with a slightly bent shaft. When you hit the ball with a putter, you putt. If you putter, it's something completely different — it means you make yourself busy without accomplishing anything huge. You might putter around the house on Saturday afternoon, filling the dishwasher and flipping through a magazine, for example. The golfing meaning comes from a Scottish word for "push or shove."