Angry zombies are simply larger, more aggressive zombies, with red eyes. They spawn occasionally among ordinary zombie spawns, but more often in magical biomes, notably Eerie biomes. Along with the usual drops, they often drop Zombie Brains (normal zombies can drop these too, but rarely) which are useful for some advanced crafting. There are also Baby Angry Zombies and Angry Villager Zombies.
Furious Zombies are even tougher, and appear to grow larger when they are hurt. They are spawned directly by Sinister nodes, similarly to a monster spawner, and with the usual limitations including light levels. Also, if there are already four Furious zombies near the node, no more will spawn until some of them leave or die. They too come in Baby and Villager forms. Note that their size increase is purely visual, and their hitbox is unchanged.