Stamp pads can contain several different kinds of ink. The five main types are: dye based, pigment based, permanent, embossing and fabric ink. Each ink serves a different purpose.
Dye based ink pads are water based and essentially leave a stain on the paper. Thus, they will run if they get wet and their colors aren't as vibrant as some other inks. On the plus side, they dry quickly and leave a very crisp image so they are good for highly detailed stamps. They are easy to clean as they leave very little residue on the stamp, although some darker colors leave a stain. Stamp cleaning solution will get a stamp clean regardless of the colors that have been used.
Pigment based inks are also water based but, unlike dye inks, they contain little particles of color. Because of this, they leave an impression that is more vibrant and more resistant to water (once dry) than their dye based counterparts. They also stay wetter longer so they are suitable for embossing. On the negative side, they can't be used on gloss stock - they smear. A stamp inked with pigment ink can usually be cleaned with a wet paper towel but the darker colors may require the use of a stamp cleaning solution.
Permanent ink is water and bleed resistant and dries quickly. It's great for stamping on unusual surfaces such as glass, wood, plastic and acetate. Most of the time, you will have to use a stamp cleaning solution to get all the ink off of the stamp.
Fabric ink is semi-permanent and, logically enough, is used primarily on fabric. Some fabric inks set automatically but some need to be heat set so, once the ink has dried, it is best to run an iron over fabric that has been stamped with fabric ink - just to be on the safe side. Stamps inked with fabric ink are also best cleaned with a stamp cleaning solution.
Embossing ink stays wet for very long time and is used only if you intend to emboss the image (For the uses of embossing ink read the article, "How To Emboss"). It comes in a variety of colors like the other inks but it is also available in a clear variety. Embossing ink is relatively sticky and should be removed with stamp cleaning solution.