In previous editions of Dungeons & Dragons each person had a number of hit points and if you ran out you were unconscious and/or dead. Most people would have no real way to restore health on their own (other than resting for months at a time) but if one of your characters was a cleric you were set for life. Clerics cast healing spells a few times a day and even if they did absolutely nothing else they'd still be mandatory because you need to be able to handle damage spikes and just keep moving from fight to fight. Maybe forcing someone to be a cleric is a problem, maybe it isn't, but it was certainly a feature of the system.
In 4th edition they changed the way healing works entirely which shakes that whole world all around. All characters still have a number of hit points which will knock you out and/or kill you when you run out, but now everyone has a second health related stat: healing surges. You get a number of these things based on class and con value and basically you can spend one at any time out of combat to restore 25% of your maximum health. So, as long as no one actually dies and everyone has at least 4 of these things left you're guaranteed to be capable of being at full health for the next fight. You can also take an action once during each fight to get the same amount of healing. If your group doesn't have a healer in 4th edition you aren't just screwed after the first fight. As long as you won and no one died you can keep playing.
Healing classes still exist in the game, though they're more hybridized than in previous editions. All of the healer classes can be built in ways such that they can do reasonable damage, or buff the party's attacks and defenses on top of the extra healing they bring. They bring the extra healing in two primary ways: by letting other people use healing surges in combat and by making those healing surges better.
Normally each person can only heal themselves once per fight, and it costs them a standard action to do so. Standard actions are pretty good and in a vacuum are way better than healing for 25% of your health. You'll use it in a pinch or if you really have nothing better to do but you're rarely happy about it. Healers can spend minor actions to let other people heal themselves with their healing surges. The healer still gets their standard action too so the party isn't losing out on damage done in this process. The healer can also hit the same person multiple times so if the enemies are threatening to spike someone down they can deal with it.
Perhaps more importantly the healer modifies the amount of healing done. This varies from class to class. I'm playing a shaman, and the basic idea is my minor action heal (which can be used twice per fight) heals a second target as well. Clerics make the heal a lot bigger on the one target. I'm not really sure what other healers do, exactly, but I'm pretty sure they all do something. This can actually be a huge difference in the amount healed compared to someone just spending their surge themselves.
As an example Aidan is playing a rogue and has 37 maximum health. His surge would heal himself for 9, though we found a magic item which raises everyone's number by 1 if they stand near Sky. So, he can spend a standard action to heal himself for 10. Or he could almost certainly hit an enemy for 20ish damage with that action instead. I took a bunch of healing related feats and items, so if I make him spend a surge instead with my healing spirit spell he's actually heal for 9+1(Sky's belt)+1(my neck)+5(my wisdom) or 16. Also, someone else would get an extra d6+1 healing which averages to 4.5. Also, everyone standing beside my pet (guaranteed to be at least one person and often 2 or 3) gets an extra 5 temporary health which will get damaged first if they get attacked. So all told we can either heal for 10 as a standard action or for ~30.5 as a minor action. It's not all in one spot and I have to modify my whole turn to position my pet properly but I get thrice the effect and still get to keep my standard action. So, yeah, healers may not be mandatory to play the game anymore but they're still pretty mandatory for being awesome.
What I want to look into, however, is how to heal up after a fight. In this case temporary hit points don't help at all and we have to commit to standing around for 5 minutes if we want to recharge my healing spells. If we're not willing to do so then healing up becomes easy... Everyone uses surges to get near full and we move on with our lives. The tricky part comes when we're willing to wait a little bit. Now I can actually heal people without costing them surges (I can land the d6+1 where I want) and surges are worth x+7 instead of x+1. The x varies from person to person, so I should try to have the person with the biggest number spend their surges and spill over to the people with the lowest number. I also want to try to bring people as close to full as I can so I don't waste the d6s or the surges. It turns out there's a feat which makes all out of combat healing dice roll maximum and I may want that feat at some point just to make the process deterministic! For now it seems like the right idea most of the time will just be to make Sky spend his surges and try to spill into Aidan since Sky heals for much more per surge (and has more surges) and Aidan has the least of both. Thinking about it a lot is unlikely to really help while I'm rolling dice. I should probably work out how many x+7s it will take to heal everyone and then see how much healing each person would need from d6+1s in order to need one fewer x+7.