Like snowflakes, no 2 pipes are exactly alike. Or at least they shouldn't be. The beauty of pipe smoking is that beyond the actual act of smoking, you define the character and personality of the utility - the pipe - itself. The smoker gets to choose the pipe design, material and look that fits his/her taste. One can even have different pipes for different occasions.
While pipes are unique, they all share the same basic components:

Each component can vary dramatically in shape, size, depth, material and color. The variances in these components is how smoking pipes are identified.
The most important component to define a smoking pipe is usually the bowl. This in combination with the material, will usually be how a particular pipe is labeled.
Common pipe bowl shape names are Dublin, Billiard, Egg, Acorn and Apple. See here for examples.
By far, the most common material for smoking pipes is Briar wood. This material comes typically from growths on the roots of the White Heath Tree (Erica arborea). The wood is very porous and very tolerant to heat which makes it ideal for smoking pipe bowls.
Another common bowl material is corncob which is very inexpensive and easy to come-by. Every pipe smoker should have a classic corncob pipe in his/her collection.
The Calabash pipe, is traditionally made from the Calabash gourd and is the smoking pipe that Sherlock Holmes used. The shape of the Calabash pipe has been mimicked and no longer is required to be made from the gourd but simply have the familiar shape. See below.
Tapered-Bit Bent Egg Smoking Pipe
Saddle-Bit Prince Smoking Pipe
Half Bent Dublin Smoking Pipe
Saddle-Bit Straight Bulldog Pipe
Long Bent Apple Pipe
Straight Poker Pipe
Traditional Calabash Smoking Pipe
But of course, the tobacco is the best part of smoking pipes!