PVP is an entirely different boat than PvE. To be perfectly honest, I'm better at it in theory than I am in practice - but that being said, I'm not bad. So here's what I do know - I'm sure others know much more. This is going to be mostly a smattering of random tidbits I've picked up through experience and conversations with other druids.
As a Balance druid, you have the advantage of your DPS capabilities boosting your healing abilities. Unfortunately, you have the disadvantage of limited crowd control and no reliable spell interrupts.
The first thing to look at is what stats you need for successful PvP. Your focus is going to shift toward spell crit, stamina, and resilience. Spell damage is still very important. Intellect remains important but is shifted back a bit - most PVP battles will be over before you have a chance to exhaust a large mana pool. Basically, the name of the game is to put out enough burst damage to kill the opponent while having enough survivability to withstand his attacks while doing so.
Most of the time in PvP, you are not going to want to be in Moonkin form. True, it gives you 5% crit. But it completely takes away your ability to heal. I know that if I get a good crit heal in, I can heal myself to full. This means that you're in essence sacrificing 5% crit for the ability to double or triple your available health. Healing in PvP > 5% crit.
Surprise is huge. Even a couple seconds of being able to do something to the enemy without retaliation is an immense advantage. One "trick" that I find successful is to start of in cat form, stealthed. This allows you to approach the enemy, find a nice position, and open up when they aren't expecting it. My 2v2 arena team is called "Spanish Inquisition" for just this reason - 2 rogues, 2 druids. As in "No one expects the Spanish Inquisition!" The added bonus is that when people are watching out for stealthed opponents, they usually don't expect that opponent to be using ranged attacks. After dropping 200 points and figuring out what the hell we were doing, we've climbed back up to about 1500 and are winning more often than we lose.
If running away is an option, that is one of the druid's strengths. If you're taking a beating, cyclone/root and travel form away. Heal up and re-engage.
Warriors - Root is your friend here. A rooted warrior can't do much. Just watch out for diminishing returns, kite in cheetah form if needed to let those wear off, and take him down from a distance. Watch out for charges & intimidating shout.
Rogues - Probably the class that I find the most annoying. In my opinion, Cloak of Shadows is overpowered with the 1 minute cooldown. That being said, if you get caught by a rogue with a cheap shot/stunlock, immediately barkskin. Then just target them and spam cyclone. Hopefully you'll survive and get it off, at which point Healing Touch + Lifebloom. Wait a half second or so and start casting roots so it lands as soon as the cyclone wears off. Back of and toss a faerie fire on him, and start DPSing from there, but be ready with your trees. As soon as he goes to Cloak of Shadows/Vanish, throw your trees up just behind where he was standing, so that they run toward you and through where he was. With any luck, they'll end up running straight through him, breaking his stealth. With vanish on cooldown, you can hopefully finish him off.
Warlocks - The initial consideration when fighting a warlock is his pet. You are generally going to see one of 3 in PvP - Succubus, Felhound, or Felguard. Either the warlock or one of the pets should be an immediate target for your roots. If the Succubus or Felhound, root the warlock - Fear only has a 20 yard range - and nuke the pet quickly. If a Felguard, then it is a Demonology lock, and you should root the pet. Immediately drop trees on the warlock, and take a second to decurse yourself if the warlock got a shot or two off. Toss a HoT up and start nuking. If the warlock gets off a drain life, cheetah and get out of the range (30-40 yards depending on spec). If you can manage to deal with the pet and stay out of fear range, you should be able to take out the lock.
Hunters - I almost always like to start by hibernating the pet. If the hunter immediately Bestial Wraths out of it, at least I made him use it early, know he's BM, and I can go cheetah and avoid as much damage as possible while BM is up. Apart from that, use instant cast HoTs to keep your health up while DPSing. If you can abuse line of sight - drop MF/IS/Trees on the hunter then run behind something - that can help a lot. Rooting the hunter can help keep him in a suitable position for a few seconds.
Priests - Holy Priests are squishy, and shouldn't be too much trouble. They may take time to kill though, because of uber healing. On the other hand, your HoTs can probably outheal their damage output. Shadow priests are like mini-warlocks. Avoid letting them heal themselves by hurting you. Trees help, and can make them at least waste their fear. Silence is the only real trick they have up their sleeve, so be prepared to cheetah-kite if you get silenced. Discipline priests are similar to holy priests, except harder to kill. Be prepared for a drawn-out fight.
Mages - Polymorph isn't a huge deal for a druid. Watch out for counterspell on heals - insta-cast heals are best. If you can, use DoTs and LOS to wear them down, by damaging them while they can't do the same to you. Apart from that, not a whole lot to worry about.
Shamans - Elemental Shamans are pretty much in the same boat you are. If possible, draw them away from their totems. They have earth shock to interrupt casting, so watch out for that. Enhancement shamans can be dealt with much like a warrior, just still be aware they can still toss a few ranged things at you and heal themselves some if need be. Resto shamans are just a royal pain to kill, but you should be able to outheal their damage as well. For all shamans, watch out for grounding totem - not a lot you can do about it, just be aware it can be there and take it into consideration.
Druids - Other balance druids are your mirror. There's really not a lot complicated in a fight between us, it's just a bunch of smacking and trying to outposition the other person. Resto druids are much like Resto shamans - you can heal through their damage, and they can heal through yours. Annoying. Feral druids you have a bit of an advantage over - both hibernate and roots work on them. Granted, they can shift out of roots, but that's 2 full seconds and a hunk of mana for them.
Paladins - The bubble is the single most annoying ability in PVP ever. 10 seconds of pure immunity in a fight is just a pain. If they're a healer, there's not much you can do to stop them from using this. If they're a fighter and they're coming after you with the bubble on, cheetah-kite them away until it wears off. Apart from that, Paladins are tactically similar to warriors, so the same considerations apply. You just can't give them as much breathing room, because they can heal themselves as well.
Finally, remember your setting. If you're doing battlegrounds, the object is not to kill the other team. It is to accomplish the objective of the battleground. Don't engage in needless fights. I've helped win AVs by running up to the horde around Stormpike, moonfiring a couple healers, and running away. Do it a few times and 8-10 horde peel off to try to kill you. Evade them, tapping them with a moonfire now and again, and you effectively cut the enemy's forces by that much for however long you can occupy them.
In arena, you absolutely have to kill the other team. Talk to your team, figure out kill orders, tactic, etc. I honestly doubt you can be an effective arena team without voice communication of some sort - use it.
I think that's it. Not very neat or pretty, but PvP isn't typically my game. Still, it's fun now and again. :D