Any mob will do. It's the player that puts (and takes) the weakness from the mob. I'd say that by level 4 or 5, every class has at least 1 weakness using ability. When you hover over an ability, if it has a red arrow next to an icon with a plus symbol in the "tooltip", then you will cause that weakness on the mob when said ability connects. If it has a red arrow without a + symbol next to it on the tooltip when you hover over an ability, then it exploits said weakness (and does extra damage).
The following are the weaknesses in the game (but you'll be able to cause/exploit no more then 3 or 4 to a class);
Afflicted
Armor Chink
Bitten
Bleeding
Blinded
Burning
Chilled
Dazed
Enraged
Flesh Rend
Inflamed
Lethargic
Mental Lapse
Mesmerized
Shaken
Soul Wracked
Staggered
Stunned
Vulnerable