The Discipline Priest in the World of Warcraft is one of the two Priestly healing specs. While Discipline is maybe not as powerful a healer as a Holy Priest spec, the Discipline Priest makes up for it with greater survivability. She can also toss around more damage and has a better ability to mitigate incoming damage on others, via Improved Shields, as well as on herself. Obviously, this means a Discipline Priest has pretty good PvP potential, among other things.
When you pick the Discipline spec at level 10 you will get certain abilities, as follows:
While leveling to 85, via questing, you will find things to be a bit slow as a Discipline priest. You can do damage, but you're not exactly a Mage or Hunter. When questing you just need to take it a bit more slowly. Remember your shields and use them. See how many opponents you can take at one time. Your shields will keep you up so that your damage spells can, evenutally, put them down. Fighting groups will be more efficient, and gain faster XP, than fighting singles. Make sure you always have enough water to regain mana as you need it. Mana potions are also a good idea and you can buy plenty from the Aution House or make your own through Alchemy.
In PvP your heals and shields will be in great demand and the Battlegrounds are a good way to level quickly. Attach yourself to a damage class, or two, keep them up against all challenges, and you'll do well. Do what it takes to help your teams win (and you'll get a lot more experience than if they lose.) You'll also gain "honor points," which you can use to buy nice gear. If you don't care for PvP though...
In the dungeons it's much the same, you'll be keeping your teams standing while they burn the opposition down. If you like the dungeons and you're fine with the group dynamics, then this is a good way to level quickly. Besides, you'll pick up better gear than you can through questing.
Some people will actually level as a Shadow spec, since the damage with that spec is so much better. One option is to dual spec (give 10 gold to your trainer, when you hit 30, to get that ability) and have Shadow as one spec and Discipline as the other. You can then quest as Shadow, if you like, and do the dungeons as Discipline.
Post level 85 your job will be mostly the same. In heroics and raids you will be casting a lot of shields. This means that you might not win the "Most Damage Healed" contest, but you will be mitigating a lot of incoming damage to your tank(s) and your team. That makes the healing jobs easier and saves a lot of mana. The Discipline Priest is more about mitigation and damage than the Holy Priest, though they are still fine healers.
Speaking of mana... in Cataclysm you generally won't have enough. Ever. Unless the fights are short you will frequently find yourself sucking fumes. This means that you will need to be careful about your spell casting. If someone blows it or is way our of place then you may have to let them die so that you have enough mana to heal/protect more important players, such as the raid tank. You can probably heal the whole group, for about 30 seconds. This is one of the big changes to priests with Cataclysm. The opposition also hits harder.
Gear can help. Try to get as much Spirit as possible on all of your gear. You will probably still have to engage in a certain amount of Triage with your spell-casting, but your mana will recover much faster than otherwise, allowing you to be more effective. Intellect is your spellspower, so you will always want more.
In PvE (dungeons, raids) you want to get gear with Intellect, Spirit, and Mastery. The other stats are fine, except for Haste. While Haste helps you to cast faster it will also help you to run out of mana faster. Mana-less Priests cast no heals. Gem and enchant your gear for Int first and Spirit second, and then Mastery if Int or Spirit are not available.
In PvP Resilience comes right after Int, and before Spirit. You will be targeted and that resilience will keep you up long enough to do something about it. So you want Int, Resilience, Spirit, and Mastery. Since your shields are so important in PvP you will find that your Mastery stat is even more important than in PvE. Get gems and enchants as you would for PvE, but consider throwing in some Resilience gems, too.
Consumables are a big help in whatever your are doing. While you might not need healing potions, mana potions are nice to have, as are Int potions. Certain foods will also buff Int or Spirit. All of the professions have a self-only enchant which will add some Int or Spirit. Tailoring, which you can use to make your own armor, has a very nice cloak enchant that will add either Spirit or Int.
Priests or all sorts are doing just fine in Cataclysm, especially since the 4.1 patch. Whether you're leveling a Priest or engaging in the end-game content you should find that the Discipline spec, and other priest specs, are all in fine shape. Now get out there, your team needs you!