Speed controls which monster gets to attack first in battle. A monster can have a speed between 1 and 3, with 3 being the fastest. If both monster have the same speed they each of a 50% change of going first.
If a monster is switched out for another monster the opposing monster will automatically get first move rendering speed meaningless. This is not the case if a monster is KO'd in this case speed does determine who goes first.
Speed is not effected by class advantage.