Before Wrath of the Lich King shook things up, arcane was largely considered to be a complementary talent tree or at best, a gimmick spec for PvP. The tree has always had excellent PvP talents such as Presence of Mind and Improved Counterspell, but has since gained impressive damage abilities, as well. In fact, Korean pro gamer Orangemarmalade was arcane-specced during the match where he pulled off a stunning 1 vs 2 win in the Intel Extreme Masters Global Finals, using such talents as Arcane Power, Incanter's Absorption and Missile Barrage to deliver enough damage to kill an opponent with almost full health in a matter of seconds.
While most players will never face an opponent possessing that level of skill, an arcane mage left to do unmitigated damage is an obscenity. You must always watch out for common tricks such as PoM-Pyro, or Presence of Mind + Pyroblast, which when empowered can sometimes kill a poorly-geared player outright or even cut down a flag-carrying tank to half. The bad news is that Presence of Mind doesn't trigger the global cooldown, so it is often macro'd to a spell, making it impossible to detect or stop. The good news is that a few mages don't macro the spell, making it a detectable, stealable, and dispellable magical buff.
If you can dispel buffs, make sure to do it on mages, for whom even the most innocuous buff such as an intellect bonus will translate to more mana and a mean crit. Removing friendly buffs such as a priest's Power Word: Fortitude should significantly impact on their health, which traditionally isn't very high to begin with. Be wary of arcane mages who have Mana Shield up, as Incanter's Absorption will buff them for their next attacks. Use AddOns to keep track of when they activate abilities such as Arcane Power, which means they're going on the offensive.
Arcane mages also have the dreaded Slow, which is a bane for all classes unable to remove debuffs. Unfortunately, this isn't even worth trinketing out of because it can easily, instantly be recast. Some classes will suffer greatly against an arcane mage's superior kiting abilities and a literal barrage of damage. This is easier said than done, but try to weather it. Interrupt, silence, snare and stun whenever possible, making sure to snare or stun at least four seconds after a Blink, as PvP arcane mages pick up Improved Blink.
Because of their numerous instant cast spells, arcane mages are extremely mobile as opposed to the other specs, and they'll move around like crazy. Don't let them. Try not to, anyway. Keep them close and deal as much damage within melee range as possible. Be prepared when an arcane mage Ice Blocks, because Prismatic Cloak means they're likely to instantly go Invisible when they cancel -- ready AoE spells to pepper the area in the hopes of catching them, or apply a DoT in the hopes they're careless enough not to macro canceling the Ice Block with Invisibility.
The fire tree is largely considered to be a PvE raiding tree, and won't be as common on the battlefield as the other two specs. That said, fire PvP has its proponents and there are more than a few PvP talents in the tree, most notably the deep 41-point talent Dragon's Breath. The talent Blazing Speed is a staple for fire mage PvP and helps them a lot against melee classes. They also have several daze mechanics and a proc-based stun. Fire mages are suicidal and deal the most insane damage when they're up close and about to die. Actually, they deal insane damage, period.
The few PvP fire mages out there will pick up Fiery Payback, which seems like a ridiculous talent except that it also usually means fire mages are encouraged to be reckless, tempted greatly to jump in the thick of combat just to proc a talent that allows them to cast Pyroblast in 1.5 secs every five seconds. That simply means that if you manage to get a fire mage below 35% health, be prepared for a take a lot of heat. Melee classes (and hunters) should also note that Fiery Payback has a disarm proc, but keep attacking anyway because the chance is low at 10%.
Fire mages perform best when wreaking havoc on a crowd. The spec shines when the mage casts Living Bomb on everyone all the time. It's expensive, but it's about the best fun a fire mage will ever have. Crowded fights such as Alterac Valley bottlenecks or Warsong Gulch flag rooms will be have players exploding left and right. Dispelling Living Bomb procs it, so there really is no way to escape it other than 'removal' mechanics such as a rogue's Cloak of Shadows. Be mindful of the debuff and your distance, as a lot of Living Bombs in an area translates to a lot of hurt. The spell is instant cast and allows fire mages to keep their DPS up while on the move.
Most fire mages will take talents in the frost tree to pick up Icy Veins, so watch out of that buff as well as Combustion. It will mean that they're ready to cast long cast spells such as Pyroblast, and are looking to build up towards Hot Streaks. Fire mages will look to unload a lot of damage to a lot of enemies by casting a succession of AoE spells, hoping for crits as more targets mean more chances to crit. Also be wary of Firestarter, which is procced by a Blast Wave and Dragon's Breath. Patch 3.3 made the talent even more deadly by making Flamestrike free in addition to being instant cast.
Generally, a measure of fire resistance will help against a fire mage. Reducing their capacity to crit will hamper their build-up time, so high Resilience works better against them than other specs. Fire mages are practically the softest of all three mage specs, so make sure to abuse that. They will not have the survivability of a frost mage nor the mobility of an arcane mage. They know this and are prepared to die, especially in the Battlegrounds, where deaths don't have a big impact.
This means that fire mages are likely to kill themselves in the process of doing their thing, which makes your job easier. The idea is to get of their way until they recklessly get their lives low enough to be easier to handle. They're like the suicide bombers of Azeroth. They're bound to die one way or the other. It's just a matter of keeping them from taking too many of your friends with them.