Myself being a mage, and having pvp'd quite a bit, I can tell you what I've found most troubling. Keeping Arena aside, as that is another matter entirely...
Rogues usually are subtlety when they pvp, so I've found that they will most likely come up from behind, start off with a stun and a few attacks.. Mages being squishy they can get us down pretty quickly, blink out of the stun, they follow up with shadow step if they are high enough, and if not, sprint and probably kidney shot to stun again and kill is that way. I find that rogues will go for stun locks on most people, but it is particularly effective against us mages, as we don't have much health to back us up.
Druids are a pain to kill. If they are bear they do a lot of damage due to our low armor and therefore low physical damage mitigation, kitties do the rogue things and get the jump on us, and do a lot of damage quickly. Having potions that increase stealth detection are very helpful to counter this. Knowing when to hit frost nova can really help, and being able to see the stealthed characters will be of great help.
Prot Warriors hurt.. and they've got spell reflect, so, it just generally sucks. If you know when to not cast, they shouldn't cause much trouble for you. Most melee classes can beat us as we have low armor and low health.
Arms/Fury warriors.. a lot of physical damage really quickly, plus hamstring, so even if we blink, we aren't going anywhere quickly. Unless you are a human or a gnome and can use your racial to get out of it, you'd have to use your pvp trinket, and then if they do it again, you're in trouble. Whirlwind and bladestorm hurt.
Ret Pallies, Repentance followed up by crusader strike, divine storm and again, melee damage against low health low armor clothies, it kills quickly. Range is your friend. But they can still stun, and even if you get them low, they bubble, we can't remove it, and its generally not a pleasant experience.
Prot Pallies, takes longer to kill us, but they have more health, so they can widdle us down before we can kill them.
Hunters, they also are ranged, so you'd want to get in close. But, their pet is where you'd need to worry about most. If you can get rid of that, dispatching the hunter shouldn't be too hard. Again though, its ranged physical damage along with ranged magic damage. We have low armor and health. I recommend presence of mind and polymorph to instant sheep a pet then nuke the hunter.
Shadowpriests, they've got silence and dots. You'll eventually die. Can't do much against silence.. Some have their fear, which also isn't nice.
Disc Priests take longer to kill as they can bubble and similar to fighting all healers.. they can widdle you down over time. So unless you can kill them, its a long and slow death.
Other Mages, some arcane mages have imp counter spell, so, as with shadow priests, can't do much against silence. In that time period burst damage will come at you must likely. Low health = death. If your opponent is another arcane mage, most likely, they'll hit up arcane power and icy veins, if they are arcane frost, followed with an arcane blast spam and arcane missiles should missile barrage proc. Pure frost, or frostfire/frostarc, will most likely use icy veins and just come at you with frostbolts or maybe the odd fireball if brainfreeze procs. Fire mages, probably will hit combustion plus icy veins if frostfire, or if fire/arc they might just hit combustion and leave it at that. In my experience, all mages might use this time... or just before engaging, use mirror image, if at 80. Each mage being different, there is a lot you might have to deal with, which is why I hate fighting other mages, as much as I hate fighting warlocks and hunters.
Warlocks. Nasty pet, felhunter that can silence and eat your buffs. They've got fear, and they've got dots. The dots, even with resilience with take out quite a bit of your health. We have no way to heal ourselves. Lots of dots, plus pet damage, silence, fears. Hunters and Warlocks are your enemies, and their pets are annoying. Warlock pets can't be sheeped though. They've also got more health, so...
That's just my advice though, from what I've encountered in my pvp experiences.