There are many many type styles available today. IT will be hard to put them in categories. Many have made attempts to classify typefaces and they usually fall into these 6 categories. Old styles letterforms have stress of rounded forms at an angle. Italic style is slanted to the right as we all know.
Transitional style has more contrast in strokes than old styles.
Modern styles have thin serifs and have a strong geometric quality. Egyptian style has heavy square or rectangular serifs. All strokes weights are usually the same San serifs have no serifs as the name implies. Not much contrast between strokes. Before we go any further, let me point out to you what a complete font includes.