No matter what an addon does (not counting the ploy ones) it is there to help you do something faster. Be it using skills, organize your items or simply getting an overview, they all want to streamline the way you go about doing things in WoW. This goes for the healing addons as well. Some healing addons do this so well in fact, they're nearly necessary to perform sufficiently enough for many raid encounters. I'm not saying playing without addons is impossible... but to most people it probably is. The human mind is only so fast and so perceptive and could definitely need some help to perform at its best in a game like WoW.
So what addons are useful for a healer? Although I write this mainly from a priests perspective, most addons work for any healing class, the only difference might be what skills they track and the like. I'll mention my favorite, healing specific addons which are great whether you heal instances or raids. Although it is more common that you need really fast reactions in a raid setting, these addons will simplify any type of healing and hopefully make it more fun!
Let's start with the overview ones. What is it a healer needs to see? Well mainly who might need some healing of course. The Blizzard standard ui does this in an ok way, which definitely works for 5 man instances all the way up to 80. But when you start doing raids, 10 mans and 25 mans, you get a hell of alot of people to keep track off. And even in instances it will come in handy to get a quick overview of who needs the most attention at any given moment. The more people around you, the more important this gets.
So what could an addon do that the Blizzard ui doesn't?
First of all it could show you who needs to be healed at all. This is especially useful when you've got 25 possible targets to tend for, but like mentioned could come in handy in any healing situations. Some addons will highlight anyone who has taken a set amount of dmg (chosen by you) and if the person is within range of you at all (so you don't try to heal something you can't reach anyway). This will allow you to focus on the ones that need a heal -now- and the ones that need a heal in a moment.
A great complement to seeing who needs a heal, is seeing exactly how much dmg someone has taken, to help you estimate how big a heal you have to throw at someone. 20% of 20.000 dmg differs greatly from 20% of 50.000 dmg. And you don't want to start doing math subtractions when someone is dying - "so 4573/34876 means...?". To me, using deficits has proven quite useful, this way I can see that person X is at -6391 hp and can decide what to do based on that information. Knowing someones max hp is less important, imo, since what you need to know is how much dmg is someone taking, and how much more dmg can they take before they die unless I heal them. If someone with 10.000 is taking 100 dmg each second he can continue doing so for quite a while before I have to heal him. If someone with 50.000 dmg is taking 10.000 dmg each second I will probably need to pay alot of healing attention to him.
I need to know how much hps my target needs to win over the dps they're taking.
Another great feature of addons is if they can show you who needs to be dispelled. Blizzard ui shows debuffs, but not in a particularly clear way, and this is again even more obvious when having to concentrate on more than 5 targets. You don't want to scan through the complete debuff list of 25 people to decide who needs a dispel/cleanse, you want to be able to make that decision within a gcd.
An addon could also help you keep better track of people who commonly need your attention, mostly the tanks. It can separate them from the others so you'll always have an easy view over what's happening to them (this is however not a feature I use myself, since I think I can give better attention to tanks when they're where everyone else are in my ui.)
There are two addons I've tried that do these things in a satisfactory way, and they both have some benefits/drawbacks compared to eachother. I used the addon X-perl for a very long time, and there are still features about it I like more than Grid, the addon I currently use. They vary somewhat in how they display the abovementioned features, but I finally chose Grid because it displays debuffs/buffs in a more visible way than X-perl. Benefits of X-perl however is that you can get nearly as much usefulness out of it but with way less hassle of configuration. Grid needs -alot- of configuration before it does what you want it to, and X-perl wasn't as difficult to get started with. I'd recommend X-perl over Grid if you're not a dedicated raider actually, or if you don't feel like a leet addon user.
I could show you some screenshots on how they look, but my ui is so messy (but that's the way I like it!) I think you get a better idea by going directly to their download pages. And since I like wowinterface more than curse, that is where I will direct you!
--> Grid (this needs alot of extra "add on" addons to get all the features though, the basic addon is pretty stripped down).
--> X-perl (Comes pretty complete, and might therefore hold features you're not interested in. These are always just to turn off of course, but it might make the addon bigger than Grid, depending on how much extras you get to Grid. Not that size matters though ;) )