The idea of the shaman+mage combo is that they're two DPS trying to kill the mob as fast as possible, but what makes it nice is that enhancement shamans tend to have lulls in their DPS as they're very bursty with Windfury and crits. The mage helps make up for these bursts by simply throwing a couple spells into the mix. It's rarely anything more than maybe a scorch or two and if necessary, fire blast. Usually the shaman is good enough to get the mob down very far in the first few seconds.
The priest+warrior was started because of how much I disliked leveling my warrior to 60 in the early days of WoW. It's honestly not very fast unless I fight more than one thing at a time (go go sweeping strikes!) in which this combo simply can last quite a long time. So essentially, if I want to maximize time on this combo, I need to literally be fighting 4+ mobs at a time. Also, a priest helps negate one factor that hurts a warrior... the time between fights (which affects your left over rage).
The paladin+warlock was more or less setup to assist both characters. One could easily use a Voidwalker as a warlock or simply chain-fear a mob until it dies, but at times, this can be more mana intensive and the voidwalker isn't necessarily the best at holding aggro and is limited against multiple mobs. The Paladin is more like a second pet for my Warlock... an enhanced Void Walker in a sense. The real benefit is for my Paladin as the warlock definitely provides DPS which is what the protadin lacks.
Only negative aspect I can say about this is sometimes you simply don't learn everything about your class. Sometimes it's figuring out how to beat a difficult situation that shows the possible amazing abilities that your class has that may not be exactly obvious.