Blade Putters
Blade putters are considered to be the most traditional putter form. Taking their visual cues from irons, the blade putter is actually a derivative of blade irons. When the game of golf was young, greens were no where near as smooth as they are today. When golfers got close to the hole they looked for a club that would keep the ball as close to the ground as possible. The blade putter was ideal because it had a short shaft, which provided the golfer with increased control. It also had very little loft which kept the ball close to the ground and got it rolling quickly. The blade putter is the ideal putter for traditionalists who want the most feel from their putting game.
Peripheral Weighted Putters
Peripheral Weighted Putters categorize some of the most popular putters on the market today. The idea behind the peripheral weighting is to expand the sweet spot of the putter so that golfers can have more consistent results despite hitting the ball in a different part of the club. This is good for professionals but even more so for amateur golfers because their club stroke is more inconsistent. The major putters included in the peripheral weighting category are the Ping Anser and the Titleist Scotty Cameron Newport. I personally play with a putter in this category and chose to because it has my preferred combination of feel and forgiveness.
Mallet Putters
Mallet Putters are designed to bring the weight of the putter behind the ball which provides the golfer with a smooth stroke. An added benefit of the weighting system is great distance control. These putters are often lacking in the feel department but usually the golfers who choose to play them don't really mind. As a general rule of thumb, I like to use these putters as a default for all sorts of putting conditions. They work great on slow and fast greens alike. The Callaway Odyssey 2-Ball putter is perhaps the most recognizable mallet putter on the market today. There are countless others as well.
Putter Faces
Besides the design of the putter head, putters come with different inserts in the faces which change the feel upon impact. Putter designers experimented for years with different materials to change the weighting of the putter. The weighting effects the putter stroke and the overall feel of the putter. Around 1995, Odyssey was the first to successfully convince the golfing world of the benefits of putting an insert into the club face. The insert can be a different material than the rest of the putter. So while your putter is mostly comprised of a very hard material, the putter face is soft and it feels nice to put with. The most well known putter insert is the white golf pall cover on the Callaway White Hot putter series. There are also rolled face putters from Teardrop that work really well to get the ball rolling quickly. I have used those putters and I would recommend them without hesitation.