Friction shifters use levers mounted on the top, down, or head tube of a bicycle frame. The levers are connected via steel cables to derailleurs, which are mechanisms that switch the chain from gear to gear. Pulling the lever puts the cable into tension or relaxation, which either pulls or releases the derailleur and pushes the chain to a different gear. Friction shifters do not have set stops in the shift point. The rider has to manually adjust the lever so that the chain is centered on the gear. Lining up the chain and gear is more difficult on this type of shifter, but it is easier to set up initially because it does not require much fine tuning. This lever system started with just two or three gears and eventually went up to five and six gears. Friction shifters were generally considered harder to use, and had problems such as unwanted shifts and noisy operation. The use of friction shifters has been eased with the invention of things such as ramped gears and idler or floating gears on the derailleurs. Image taken from www.tearsforgears.com/2005/11/campagnolo-syncro-ii-shifters.html
Index shifters are not a separate type of shifter but an innovation in shifting technology. Indexed shifters can be levers like friction shifters, or handlebar mounted triggers or twisters. These shifters function differently than friction shifters in that they have set shift points for each gear. This means that each gear has a set position on the shifter. Once in position, the position cannot be fine tuned by using the shifter. The position of each preset shift point is determined by setup and tuning of the derailleurs. The positions need to be set perfectly from a static position since they are not adjustable while being ridden. This type of operation is preferred over friction shifters because of the ease of shifting while riding. Major drawbacks include a more difficult set up and incompatibility with other brands. Most index shifters work with specific parts, unlike friction shifters which worked with almost any part brand. Image taken from http://store.airbomb.com/Items.asp?Cc=589-T-367
Twist-Style Shifters Twist-style shifters are located on the handle bars and are operated by turning the handle bars. They work by moving a derailleur in and out which moves the chain on the gears. Most bicycles have two shifters to change two different gears. The location of the shifter makes it convenient as you don’t need to move your hands to shift gears.
Image taken from http://en.wikipedia.org/wiki/Shifter_(bicycle_part)
Lever Shifters Lever style shifters are also located on the handle bars of the bicycle. They have two levers one to shift up, one to shift down. Like the twist shifters, there are usually two for two separate gears. Often located with the levers is the brake which is convenient while riding.
Image taken from http://www.amazon.com/b?ie=UTF8&node=3404951
Both the twist style and lever shifters were each developed for the index shifters which allow you to shift directly into gear as opposed to friction shifters
Brifter or dual control levers
While index shifters were an improvement to friction shifters the rider still lost some riding stability when shifting gears. This issue was resolved with the introduction of the Brifter. The system combines an index shifter into each brake lever and for this reason became known as a Brifter. This system allows the rider to shift quickly and accurately without sacrificing stability which is important during a race. Brifters are often combined with cassette modifications which allow the chain to transfer power during a shift which utilizes the Brifters rapid shifting capabilities. The downside to Brifters are their cost and complexity. Due to the large number of moving parts they can be less reliable especially if the correct cable tension is not maintained. Over time cables stretch and require routine tightening to ensure adequate performance. Finally some brands are almost impossible to be rebuilt; Shimano STI levers in particular are notorious for this resulting in a large expense to be replaced if one fails.
image taken from http://www.spadout.com/wiki/images/Brifter.html