"Zoo" is basically a low cost minion deck in which you spam out as many minions to fill the board as you can (making a "zoo" on the screen). Typically seen with Warlock due to card draw, but sometimes Rogue and Paladin as well. Usually involves having knife-jugglers to get the extra damage when you play a new minion, and will include some minions that spawn other minions when they get damaged/or spell cast/etc (as well as things like Stormwind Champion, Abusive Sargant, timber wolf, etc., that give bonuses to other minions).
Handlock is mainly a Warlock deck (hence the "lock"), but a couple others can play it as well (druid can do it pretty well). It involves cards that have bonuses for having other cards in your hand like twilight drake, and almost all the giants. Basically strategy is to get 8-10 cards in your hand and then play things like a twilight drake such that it will have 8-10 health and be pretty powerful against anyone who doesn't have a silence, and once your character gets damaged enough, things like molten giant are free, and you find yourself facing 3-4 giants on the table as well as with 2 of them getting taunted by a Defender of Argus, and possibly windfury/shielded by enhance-o. And typically once the warlock has taken damage down into the 10's, he will play their class legendary that replaces their character (or similar legendary).