This achievement is unlocked once you have defeated your first monster. This must be done in survival mode. You cannot do this on peaceful.
Monsters are aggresive mobs, and only spawn during the night time (on the surface) or in dark places such as caves and unlit homes.
You can kill them by dealing damage to them, this can be done by pressing
or the left mouse button no matter what is in your hand. However, to do this faster and easier make a sword and hold it when you attack, this will increase the damage you do.
To make this easier you can set the difficulty to easy using the options menu.
This picture shows some mobs: