Serif/Roman
Serif or Roman typefaces are characterized by the serifs on the ends of the letters.
Historical/Celtic
Celtic typefaces (Manno, Privitera Gloekler, Reilly) are characterized by their roundedness and triangular ascender tops. Yu and Gilbert are historical typefaces.
Script
Script typefaces look like handwriting.
Sans-Serif/Block
Sans-serif or block typefaces are characterized by the absence of serifs and consistent stroke weights.
Blackletter/Old English
Blackletter or Old English typefaces are characterized by the extra embellished first initials and calligraphic flourishes.