Marking pens (or markers) can contain several different kinds of ink. The four main types are: water based, permanent, embossing and fabric ink. Each ink serves a different purpose.
Water based markers are the ones most often used by rubber stampers and come in a wide variety of colors.When used on glossy paper, water based markers produce an perfectly even finish with no overlapping of color but they tend to apppear a liitle faded when dry. On matte paper, the color is vibrant and clear but the ink overlaps especially when using darker colors. Water based markers are easy to clean as they leave very little residue on the stamp. A wet paper towel is usually enough to get a stamp clean if it has been inked with a water based marker. Two of the most popular brand names are Marvy and Tombow. They both provide two different tip widths (a large brush shaped tip and a small detailing tip) in each marker. Their differences are primarily in the size and rigidness of the tips so try them out to see which marker feels best to you. There are many less expensive ones that work as well but these tend to have only one tip per marker.
Permanent ink markers are essentially the "Magic Markers" that most of us grew up with. They have a chemical base and are water and bleed resistant. They also come in a wide variety of colors and, when used properly, tend to overlap less than water based markers. A stamp inked with a permanent marker must be cleaned a stamp cleaning solution.
Fabric markers are semi-permanent and, logically enough, are used primarily on fabric although they work well and eliminate overlapping when used on matte paper. Some fabric markers 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 a fabric marker - just to be on the safe side. Stamps inked with fabric markers are also best cleaned with a stamp cleaning solution.
As both water based and permanent markers dry very quickly, you will need an embossing marker if you want to emboss an image (see, "How To Emboss"). They come in a limited variety of colors compared to other markers but they are also available in a clear variety. Embossing markers use a relatively sticky ink and should be removed with stamp cleaning solution.