Werewolf pack's enforcers are alpha werewolves who have been assigned the task of making sure the Ulfric's wishes and orders are carried out. They also provide protection for the Ulfric and anyone or anything the Ulfric wishes to be protected, although they cannot interfere with legitimate fights of succession.
Enforcers presumably need to be strong and capable fighters as well as dominant personalities to be able to carry out their duties. At the same time they should be unerringly loyal, subservient enough to bow to the Ulfric's will, and willing to take a bullet for him. Consequently, when the Ulfric is defeated in a fight for the leadership of the pack, the new Ulfric traditionally kills the previous enforcers as well. This keeps them from being a threat to the new establishment, both physically and in their ability to influence and order around the weaker pack members.
The primary and secondary enforcers have titles Sköll and Hati, but there might be lower level enforcers as well (hasn't been stated). The enforcers we have seen the most, Jamil and Shang-Da, are both very dedicated to Richard, their Ulfric.