The immensely important role of healers in PvP usually means that you will be responsible for main healing, outlasting burst attacks by enemy rogues and keeping your priests alive. If you are lucky, you are on stealther missions with a team of rogues. In case your team has a significant healer surplus and the other team is low on healers, you might want to volunteer to disturb their healers with cat form attacks, bash and feral charge.
Warsong Gulch
From level 30 on you have an instant cast 40% speed buff (travel form), making you one of the best flag running classes. With proper PvP gear that bonus can be as high as 55%. Additionally your shape-shifting ability can get rid of any snares you might have on you. In advanced groups Warsong is more similar to grouped PvP where people stick together and move as one team. This section deals more with smaller (duo and trio flag running teams) which are common for pick-up groups.

Warsong Gulch is a druid's playground; it's like it was made for the druid. You can heal your flag carrier, root an enemy flag carrier, stealth into base to grab the enemy

flag, use Dash and Travel Form to cross the field as a flag carrier, switch forms to get rid of some movement restricting effects, use Dire Bear form to shrug off damage (pop a Frenzied Regeneration for a free heal) and keep running, in short, played well, a druid can single-handedly win WSG.
?Priests can AoE fear, cast shield and renew, giving you an additional HP bonus. Excellent team, but it’s rare that your team can spare two healers for a duo attack.
?Warriors can cast AoE fear (need to build up some rage though first) and Hamstring to help you get away. An alternative is to have the Warrior take the flag and make a run while you wait stealthed. Once the warrior dies and loses the flag you pick it up and start the real run.
?Mages can lockdown flocks of players with frost nova and use polymorph to take out especially dangerous enemies. A rather nasty strategy is to have the mage spam instant AoEs to lag slower computers down.
?Paladins – similar to the Warrior fake-running strategy. They take ages to take down. If your Paladin is really well equipped, have him run through the tunnel while you wait half way down healing him. Paladins can get rid of snare effects comparably easy and might be able to make it outside with only minimal support even against overwhelming numbers (given the right equipment).
?Shamans can use totems to slow down enemies and frostshock dangerous defenders.
?Warlocks might get a Howl of Terror of while their Succubus seduces your closest pursuer.
?Rogues – the fastest flag runners. Properly specced they have two 15 seconds 70% sprints which they can usually follow up with a trinket. In other words: If they get the speed buff in the tunnel they can sprint to their own base faster than regular mounts. Usually druids go for the fake attack with larger defender groups, with smaller groups it’s the rogue’s job to fake (since there will be more defenders fighting on the field).
?Hunters can frost trap the tunnel and concussion shot the closes pursuers (they should wait in the tunnel). Alternatively they can distract defenders by firing down from the roof.

Druids make interesting partners, you can either have one do a fake run, or work as a duo (if the odds aren’t too overwhelming) to reach your support group.
General Advice
Take a look at the battleground before grabbing the flag – can you make a fast exit by feral charging a bad (or for you: well) positioned defender? Which of the defenders have used their special abilities (Hamstring, Death Coil…) already and aren’t as dangerous anymore?
If you can’t get around an especially persistent flag guard, open with pounce before you grab the flag – that buys you another two seconds. Alternatively shift to bear and bash.
Be sure you have track humanoids on in kitty form. You'll know right where everyone is (except for stealthers).
If you REALLY want the win and are willing to bear the costs, use potions and food buffs. The Free Action Potion combined with the kitty's Dash and travel form will almost always get you across the field untouched to your base.
Arathi Basin
If you are facing a zerging team, set up a duo with a rogue and capture nodes with only one or two defenders (three if your rogue knows what he is doing). Work together to sap damage dealers and kill healers.
On the move watch out for stealthing classes. If you see a single stealther riding towards one of your bases, throw in a Faerie Fire – that will cost him at least 30 seconds – causing him either to ride back or wait and sit it out.
While defending wait stealthed in cat form. Do not stand too close to the flag or a scouting rogue might discover you before you see him. If you are facing overwhelming numbers, cast HoTs, DoTs and switch into bear and hopefully last till help arrives. Use frenzied regeneration and NS (if you have it) to heal yourself and swipe as many targets as possible around the flag.
Moonspam level 1 is the easiest way to prevent flag captures – remember that hurricane won’t prevent flag caps.

Don't fight out in the roads! Fight near a flag. Let the enemy go, don't be drawn away from your flag.
Do the kitty stealth thing: when defending a flag or moving to take a flag, stealth in and take out the cloth wearers; that's a feral druid's best role. Get the healers first. Pounce, to stun them, get a couple of combo points on them, Maim to shut down the spell they're casting on you, then Mangle and Shred leading to a Ferocious Bit to finish them off.
Cap the flag the instant you can to prevent the enemy from rezzing at that base.
Alterac Valley
Your services as Tank, Healer, or DPS are are vital in Alterac Valley. This battleground continues to evolve as Blizzard makes small changes in an attempt to fix perceived balance issues. But as a druid, you can still tank the enemy NPCs, heal your fellow players, and stealth in to enemy towers to try and steal them from right under the enemy's nose.
Feral Druids
Your bear (at the top level for the BG range) can tank the enemy general and captain.
Your kitty can take out healers, pick off enemy players engaged in battling your teammates, and stealth into towers to pick them off. The kitty is a great tower guard, stealthing to defend, keeping an eye on incoming enemies with Detect Humanoids, and using Pounce and Maim to slow down enemy attackers.
Resto Druids
Your healing and crowd control are vital to any victory. Keeping the tank up when hitting the general and captain is one of your primary services. You stand behind the constantly shifting battle lines healing your teammates, extending any attack. (Ctrl-Tab to cycle through nearby allies, slap a Lifebloom and a Rejuvenation on anyone with reduced health.)
When defending a grave yard or base, use Roots and Cyclone to control enemy attackers.
Balance Druids
You job is to crowd control, buff, and mostly to do damage. You a great big target in Moonkin or caster form, so stay back behind the meat shields.
Heal if you have no better use for your mana at the time.

Alternatively you can be employed to hunt enemy farmers (who hunt rams/wolves) – plenty of 1on1s, and you keep them from using their special abilities.
Eye of the Storm
This battleground is a combination of Warsong Gulch and Arathi Basin. There are four towers to capture, each tower owned gives a flow of points until one teams reaches 2,000. A flag can be captured in the middle of the field and run to a owned tower for bonus points.
Druids do as well here as they do in WSG and AB. The same strategies apply, for the most part.
The trouble with EotS is that everyone on the field wants to run the flag. This is a losing strategy. Owning two (or more) of four towers and capturing the flag is a winning strategy. This requires most of the team capturing/defending towers, and only 3 or 4 players contending for the flag.
Druids should stealth and protect a tower. Druids also make good roamers, with Detect Humaniods on you have a shot at stealing a tower with no defenders. And, of course, druids make great flag runners, using Dash, Travel Form, and Bear to capture a flag.