The levels are randomly generated, so it is totally random whether or not a monster will show up. This achievement is cumulative, so you don't need to kill 3 monsters in one level or the same playthrough.
In order to defeat a monster, you need a pickaxe, machete (machetes don't go away when used, but do disappear at the end of a level), arrow, or dynamite. Each weapon or tool is one only, except for the machete as that can be used multiple times, but only in the same level it was picked up in.
There are snakes, bats, skeletons (and possibly more), but you should be easily able to see what is a monster.