From Battlefield to World of Warcraft, healing characters are arguably the most important characters in any good team. They sacrifice their own personal health and damage to keep the entire team healed, mitigating the effects of a war of attrition. Without them, even the best damage dealers and tanks will die eventually. This maxim is especially true in Overwatch, where a good healer can mean the difference between maintaining a constant push on an objective and being forced to take the walk of shame from the spawn point.
Unfortunately, this does mean that you will have to rely on your team to protect you; a fact that seems to be lost to certain players in Overwatch.
As the most traditional healer of Overwatch, Mercy specializes in healing one person at a time. As such, she can keep most Tanks alive even if they are taking an otherwise obscene amount of damage. On top of that, her ultimate ability lets her resurrect any dead teammates within roughly 20 to 30 feet of her, meaning that she can instantly turn a disaster into a salvageable situation. She can also provide a roughly 30% damage boost for any character, making her a deadly combination with almost any character that has a naturally high rate of fire or damage.
Unfortunately, she is also the Support character that relies upon her teammates the most. While her Caduceus blaster can do respectable amounts of damage (at 20 damage per body shot), it is entirely a last resort weapon for situations where your team cannot, or will not pay attention to anyone that is attacking you. She can also fly to any teammate, but it works on the same basis as her healing beam—she must have line of sight on a teammate. Realistically, this means that playing as Mercy might force you to choose who to heal and who to leave to die; you can't save people from running off and getting themselves killed after all, and if someone is consistently ignoring threats to you, then it might be best to let them die rather than risking your own life.
Compared to the rest of Overwatch's Support characters, Lucio is possibly the easiest to play as. Unlike the other healers, he doesn't have to maintain a line of sight on a teammate to heal them given his passive Area of Effect healing ability. If needed, he can also switch from a healing AOE to a speed boost AOE, making it easier for the team to get to the objective while making it harder to hit him. At worst, all you have to concern yourself with is keeping track of how many teammates are within range of your AOE burst ability and when to use your ultimate to push a point or negate an enemy ultimate.
The only real downside to playing as Lucio is that his primary weapon is somewhat lackluster at medium range, and utterly useless at long range. Unless whatever you're shooting at is completely oblivious, chances are that anyone who has eyes will be able to move slightly to the side, making you miss your shots and potentially giving them a chance to mow you down.
Filling a role somewhere between Mercy and Lucio is Zenyatta. Of the healers, Zenyatta does the most damage and has the easiest projectiles to aim, making him surprisingly deadly to other characters. However, he is a true glass cannon, despite being a mechanical being; with only 150 health, a Widowmaker can instantly kill you with just a single bodyshot, to say nothing of what would happen if a Reaper or Tracer sneaks up on you. Fortunately, his Orb of Discord increases the amount of damage that an enemy takes by 50%, making him very useful at helping his teammates kill high priority targets.
Unfortunately, the rest of his abilities do not encourage a very passive playstyle. To heal people, you must throw an Orb of Harmony at a person, but it only lasts for three seconds if you lose your line of sight to the teammate. Similarly, his Orb of Discord will automatically expire if you lose sight of the enemy, making Zenyatta a very high risk, high reward kind of character.
Generally speaking, most of Overwatch's support characters are capable of healing teammates. This makes Symmetra the odd one out, seeing as how she has no healing ability whatsoever. The closest thing to healing that she has is her ability to provide a meager 25 shields to everyone on your team, which is barely enough to account for an extra shot or two. Needless to say, she cannot keep your teammates alive, but she can make it so that when your teammates do die, they can re-enter the fight much faster. Her (fairly fast charging) ultimate lets her make a teleporter at any position, which confers an obvious advantage to naturally slow characters (but only six teammates can use the teleporter before it destroys itself).
Fortunately for her, Symmetra does have the ability to place up to six very small, very annoying laser turrets down. While they lack the raw firepower and durability of Torbjorn's turrets, they can be placed anywhere. That means that you can put them under an arch, line a doorframe with turrets, or place them in other annoying locations to slow down enemies. Her primary weapon is no different, letting her microwave someone to death at close range, but it is incredibly useless at any range greater than 5-10 feet away. On the bright side, her weapon's alternative fire mode can go through things like Reinhardt's barrier, so she can contribute at long range in that regard.