There are six main categories of type: Old Style, Modern, Slab Serif, Sans Serif, Script and Decorative.
Old Style
These serif fonts have a noticeable thick to thin transition. They have a slanted serif, and diagonal stress to mimic the angle of the old pens.
Modern
These serif fonts have a perpendicular serif line, and a vertical stress (meaning it's thicker on the vertical strokes). It no longer looks hand-drawn, and is more extreme in the thick and thin contrast.
Slab Serif
This serif style came along with the industrial revolution and is thicker all around, and more readable than the modern or old style serifs.
Sans Serif
Usually monoweight–no thick/thin transitions–and no serifs on the ends of the strokes.
Script
Hand lettered, calligraphy, cursive. Use sparingly.
Decorative
Very expressive and distinctive. Not for text blocks. Use sparingly.