Modern is what was meant to replace OLDSTYLE, it goes from thin to thick and has horizontal serifs an example of modern is bodoni, this font has horizontal serifs and goes from thick to thin
Slab serif is used for advertising because it is easy to read from far away, an example would be Rockwell because when typing it the lowercases have a serif that looks like a slab.
Script is a font that looks as if it was handwritten, they dominate designs and are easy to spot, an example is apple chancery because it looks handwritten
Decorative fonts are fun and very distinctive they are great for expressing moods, an example would be chalkduster because it looks as if someone wrote it with chalk.