Definition of monster
A monster is a hand that is very strong. Monster is not an official term, but a common parlance expression. Generally speaking, we'll want to "slowplay" our monsters, i.e. play them passively so that your opponent stays hooked. For example, if we have a pair of jacks in our hand and the flop comes 2-2-J, we have a monster, a hand that is almost unbeatable (except against a four of a kind, of course).
Should you play your monsters fast or slow?
The answer depends on the texture of the flop or board. If the board is dry, it's best to play passively. If the board is dynamic, offering several draws, it's best to play fast, even with a monster! Let's look at 2 examples. You have 99 in your hand on 9-J-Q. You certainly have a monster, but the board is dynamic. And if there's also a flush draw, you'll want to play this flop as quickly as possible. Bet big and raise big. When faced with a raise, over-raise! In another situation, you have 55 in your hand on 2-5-5. You have the best possible hand, which obviously qualifies as a monster! Draws are almost non-existent, as the board is rainbow (spades, clubs and hearts), so there are no color draws. On this type of board, play slowly! Checks and calls are preferable.