If it's a healer, you ain't Han Solo
Remember that healers are balanced to be able to survive at least one DPSer. If they couldn't stay up versus one (or two) DPS players, there wouldn't be much point in having healers around. That being said, if you're one, lonesome fighter waling away on a healer, settle in for a long, painful ride. Because you're not killing that healer by yourself.
As soon as you realize you're up against a healer (you can tell by the heals), shout out to your team that you need help. I don't care if you're Rogue McNorris, Chuck's long-lost stabby master. Get help; get it faster.
Having a healer near the FC is no better
Of course, having a healer on the FC (in lieu of the FC's being a healer) isn't really better. In some ways, it's worse, since peeling off the FC in favor of roughing up the healer leaves the FC untouched. Again, if you're by yourself, call for help.
Let's just get this out of the way. Healers throw some darn amazing heals in the Battlegrounds, and you're not going to kill a flag carrier or healer by yourself. Sure, it's possible for Rogue McNorris or a competent death knight, but as a general rule, if you see a healer, you're going to need help.
Go where the FC will be, not where it is
Healer issues out of the way, the first thing you need to do is catch up with the flag carrier. If you're a ranged character, you still need to get inside firing range. To borrow a phrase from Wayne Gretzky, you need to get where the flag carrier will be, not where it is right now. That means two things.
First, if you think you can close the distance, lead your target a little. Run to a point ahead of their path. That lets you close the distance faster instead of trying to catch up. That plan isn't terribly deep; you probably do it already.
Second, and more importantly, you can try to head the flag carrier off at the pass. Does it seem likely the FC will try to get up the tunnel? Is he heading for Mage Tower? Getting yourself a little ahead of the FC's route gives you extra time to gun them down while he's running toward you, instead.
Of course, you could be wrong about which way the FC is going, but that's the nature of the game.
Slow him down
It's entirely possible for you to kill a non-healer flag carrier by your lonesome, especially if that flag carrier doesn't have any protection. But since the goal of the opposing team should be to protect their FC, chances are that someone's going to defend your target. If you can't get the kill quickly, what do you do?
You slow them down. Use crowd control, snares, and anything else at your disposal. Your goal here is to delay the FC's movement long enough for your own team to show up and provide backup. Slowing the enemy flag carrier is more effective in most cases than just laying down the damage.
It goes without saying at this point that you probably need to kill the enemy healer first. If you're slowing the FC, that gives everyone else on your team the chance to kill the healer. See how this works? Slow them down, wait for backup.
Master the plan
To put the entire strategy for killing an FC in very simple terms: Slow the flag carrier down to give your team a chance to catch up with you. If there's a healer there, definitely make sure you have help coming.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to learning how to tank, getting up to speed for heroics and even how to win Tol Barad.