If you're looking for a writing implement, you have a bunch of options. Pens never lose their tips, but you can't erase all varieties of them. Pencils, on the other hand, need to be sharpened for use. The mechanical and propelling pencil are two different attempts to solve that issue.
A mechanical pencil is usually made of plastic and has an advancing mechanism built in to push the rods of graphite through the writing tip. The other end of the pencil is usually hollow with a rubber stopper, which both acts as an eraser and holds the extra graphite in the body of the pencil. The graphite slides into the mechanism from the chamber, and when the button is pressed, the mechanism slides a small amount forward.
Propelling pencils are also made of plastic but lack the mechanics that make up a mechanical pencil. They instead use plastic pods filled with graphite tips. The pencil body itself is a hollow plastic, pencil-shaped tube. The pods have a thick, pointed piece of graphite, usually about 1/2 inch long, and a hollow rear end. The pods are each stacked in the body of the pencil. When the tip of one pod gets too dull, you can pop it out by pushing on the last pod in the pencil. It will force the rest of the pods to advance, letting you put the depleted pod in the tail end while the fresh one is now usable.
Mechanical pencils come in a wide variety of styles and shapes. Most of them will either have a top or side button that can be pressed to advance the graphite, but some use a twisting mechanism instead. The most common varieties are made of plastic, although metal ones exist as well. Propelling pencils, on the other hand, are pretty standard, although they may come with a plastic cap to cover the tip when not in use.
Mechanical pencils do have a potential drawback: They end up wasting a lot of graphite, as the mechanism needs a certain amount of graphite to function. When a rod becomes too small, it will stop advancing. Even more unfortunately, it's easy to break a rod when it's only a few millimeters thick.
Propelling pencils, on the other hand, are usually decorated with comparatively juvenile designs, and the tip is fairly hard to manage. It starts off extremely sharp when the pod is first advanced, but after a few minutes of use often either breaks or dulls. Once dull, it's possible to resharpen it, but you'll have to use the lead to make it sharp again.
The mechanical pencil and propelling pencils are both suitable writing tools for situations where a pen won't cut it. If you need to be able to erase something, a pencil of any kind will work. But, if you want a thin, neat tip without having to sharpen it every few minutes, the mechanical or propelling pencil are ideal for your work.