There are 3 basic classifications of Insurgent groups:
Sadaam/Baathist Loyalist : (aka the FRL's, "Former Regime Loyalists")
Though Saddam himself is pretty much out of the picture, his party, the Baathists, along with the Sunnis, base thier gripes on the fact that they used to have control, but now only have a 20% share of the national vote, at best. They also dont like the fact that they dont live in the oil rich regions, therefore they want those revenues back, while those regions with oil want to keep the revenue local. The sunnis are willing to resort to intimidation and terrorism to get it. They are the richest of the 3 major regions of Iraq, and want to keep it that way.
Al Queda & other nutjob groups - (aka "Islamic Revivalists")
a) These are the guys who were brainwashed by clerics since age 3 about how killing the infidel non muslim will get you straight to heaven, along with 70 virgins and yada yada yada. They will travel half way around the world to blow themseleves up near an american. The evidence of brain washing is apparent when they are told to sever contact with thier families and friends, and are restricted to speaking only to thier fellow "trainees" they usually have to train in camps as to avoid outside influence.
b) These are your basic mercenaries, they get paid way more money to shoot at Americans and to build road side bombs than pretty much any other job available. The fundies have money, and when brainwashing takes too long, giving someone a few thousand bucks to raise some hell is a pretty effective method.
Both A & B of the Revivalists are where you are likely to find the most foreign figters.
Tribal Groups:
These groups basically act out of tribal loyalty, which in Iraq is stronger than loyalty to your own family. If someone in a tribe gets killed by an american, the whole tribe is bound to "even the score" regardless of what thier fellow tribesman may have done to get himself killed. The tribes have also been known to "play the field" with Americans, they will set off a bomb, and then act as informants, and blame it on a rival tribe, hoping the rival tribes gets raided and a firefight erupts. There are about 150 tribes that make up the Iraqi populace.
The composition of the overall insurgency is in a constant flux. The coalition forces usually focus on the largest one, marginalize it, then change focus to the next biggest threat. In late 2003, FRL's were the largest, most powerful and most capable, while foreign fighters were almost non existent, a year later it was like the total opposite. Foreign fighters became proficient at securing safe houses from which to operate, while saddam loyalists had largely been wiped out or given up (aka lost thier funding) Then once the coalition focused in on these safe houses, they forced much of the foriegn fighters to the west. Which was pretty stupid if you ask me, a terrorists best friend is nearby collateral damage. Once they were out in the open, their camps were getting creamed by US raids. Recent patterns suggest they are trying to establish staging points within the cities again.
A recent New York Times study of the data reveals that 80% of attacks classified as "insurgency" had no political motivation at all, they were simply criminal in nature, ransom kidnappings, hijackings, lootings, etc. This just shows that most people are just taking advantage of the lack of a widespread police force.