Before I start, I recommend that any WoW player who is unfamiliar with addons read up on the idea itself. Addons can do add any number of things to your game, from making your user interface (or UI) more customizable to measuring your DPS to giving you a heads-up when your pizza is ready. Installation requires you to be able to navigate your computer's file directories, and they take varying levels of fiddling to configure, but if you don't already use them they can change your gameplay completely.
Anyone who regularly plays in a group situation will want to have a few particular addons installed. Omen Threat Meter, Recount, and Deadly Boss Mods or variations on them are all very basic, and they're all covered extensively elsewhere so I'm not going to go into detail on them here. They're also just as useful no matter what class you play (except possibly the first two for healers), so there's nothing very Death Knight-specific to discuss. Just remember that tanks want to be at the top of Omen, and DPSers want to be at the top of Recount! Though sometimes DPSers can get confused and think they want to be at the top of both.
For those already very familiar with addons, there are some I've found especially useful for Death Knights. Even a beginning DK can get a great deal of use from DeathKnight.info Runes/DeathKnight.info Diseases from the DeathKnight.info community. This is technically two addons, but they come packaged together and both are extremely useful. DKi Runes is a bar that you can move to anywhere on your screen that gives a graphical representation of your rune cooldowns as well as two other configurable bars that can be set to your HP or Runic Power. This lets you move your rune bar to somewhere more obvious than your character frame, and it looks really awesome to boot. For beginning DKs it's perfect for forming an awareness of your cooldowns, and even experienced DKs can benefit from tracking them more easily.
DKi Diseases fills a tracking function as well, in this case tracking your diseases. This addon creates movable, configurable bars to show how much longer your diseases will be up on your main target, with another underlay (or overlay) showing a timer for diseases spread to other targets by Pestilence. I can't stress enough how important it is for Death Knights to keep their diseases up! I've personally set mine to display right under the target frame below the other DoTs so I can glance up and see when I need to refresh them.
While it's not specific to our class, as a Frost DK I've found Power Auras Classic exceptional for making sure I get the most of my procs. It's an extremely fiddly little addon, and it may take a bit of tweaking to get it to do exactly what you want, but fortunately there's an entire wiki devoted to it. I use it to give me a heads-up when Killing Machine or Rime/Freezing Fog have procced as well as when my RP is getting too high. I have also tied the latter to a sound file of the Corrupted Ashbringer saying "Kill them all," because as we all know, runeblades are evil bastards. It's also good for letting you know if you're in the wrong presence, letting you know when Horn of Winter is no longer up, and any other warning you can think of. I've poked at it a little on some of my other characters, but my Death Knight is the one for whom I've found the most use for it.
I know there are other DK-specific addons out there I haven't tried, so feel free to comment with a testimonial on your favorite!