There are many different classifications and sub-classification of typefaces, but the most common two types you will hear of are: Serif and Sans-serif.
In typography, serifs are semi-structural details on the ends of some of the strokes that make up letters and symbols. A typeface that has serifs is called a serif typeface (or seriffed typeface).
Some typography sources refer to sans-serif typefaces as "grotesque" (in German "grotesk") or "Gothic," and serif types as "Roman."
In typography, a sans-serif or sans serif typeface is one that does not have the small features called "serifs" at the end of strokes. The term comes from the Latin word sans, meaning "without".