What role is a "Heavy" supposed to fill?

The main purpose of heavies, early on, is a panic button: if you pull too many sectoids, thin men, and floaters all at once, you can nuke a fair number of them with the heavy's missile, with a near 100% chance to hit (the tooltip says 90%, but most of the 10% misses land very close to the target spot, and usually hit everything you were aiming at). It's also quite effective against mutons hiding behind cars, vans, or trucks, since the secondary vehicle explosion happens immediately.

They're also great when you have tough enemies hiding behind destructible cover, and want to improve others' aim by blowing it up.

Mid and late game, they're your primary means of taking out deadly robotic units via shredder rockets and/or heat ammo.

Often, when you build your first heavy, you'll be tempted to take the wrong skills, because some of the worthless ones sound pretty good...I know I took almost every wrong skill on my first one, so I think a big part of knowing what role a heavy should play is knowing how to build one:

Heavy Skills

At corporal, Bullet Swarm makes them very effective when you have a swarm of enemies advancing on you, and helps to make up for the fact that heavies have the worst aim of any XCOM soldiers. Holotargeting is usually not as useful: a hit from a heavy is worth more than a hit from any other class (except sniper), more shots generally results in more damage than a measly 10% aim bonus against one unit, and there are better options (Bullet Swarm, Shredder Rocket, the assault's Run and Gun + Rapid Fire, the sniper's Head Shot) when you need to quickly focus something down.

At sergeant, Shredder Rocket gives you a second panic button against large groups of weak enemies, and an excellent way to deal with deadly robotic units: hit it/them with a shredder, and the damage amplification will help the rest of your crew take it/them out quickly. Suppression is generally not as useful, in part because an alien that's dead is less likely to hit you than an alien with -30% aim, in part because it burns too much ammunition: it requires two of your heavy's three ammo to use the skill, meaning you lose the option of using Bullet Swarm on the following turn, and you can't keep an enemy suppressed for more than one turn. It often leaves you out of ammo in deep trouble. On the other hand, Shredder Rocket gives you an additional option when you're out of ammo. The aim penalty on the reaction shot (if it happens) is also pretty big for a heavy, whose aim isn't very good to start with.

Compare that to a support's Rifle Suppression, which a support can use two turns in a row even before Ammo Conservation, which doesn't prevent taking two shots in a turn (since a support can't anyway), and remember the support's aim is second only to the sniper, so the reaction shot penalty isn't as big a deal, which makes the support usually the better option for suppression, if you like to use it.

At Lieutienant, HEAT Ammo is the clear winner over Rapid Reaction, and the number one skill that makes heavies worthwhile: do you want two shots of reaction fire, given a heavy's low chance of connecting on a reaction shot? With the second shot fired only if the first one hits?? To boost the rate at which your heavy runs out of ammo??? Or do you want double damage against the deadliest enemies in the game, and the ability to take one out single-handed if both shots from bullet swarm hit?

At Captain, both choices are an option: grenades are another guaranteed hit area-effect weapon, which is useful given the heavy's poor aim, and, like rockets, they're useful for destroying cover and exploding vehicles that aliens are hiding behind for bonus damage. If you like your heavies to carry grenades as their accessory, it's the obvious choice.

On the other hand, two extra tiles' radius on your rockets from Danger Zone makes it easier to get large numbers of enemies in the blast radius, and is the only option if you prefer to use a scope to boost the heavy's poor aim, or anything other than a grenade.

The bonus radius on suppression would almost make for an argument to consider suppression, but in my experience, aliens rarely end their turn within two tiles of another alien, so suppression remains single-target most of the time. (If you could aim the diameter five area of suppression fire, instead of having to center it on a specific alien, that might be different, as you'd often be able to suppress three instead of just one if you could center the suppression fire on open space.)

At Colonel, a third rocket is generally more useful than +2 damage on the two rockets you already have.

Heavies are your primary damage dealers.

• Their exclusive heavy weapons are simply better than the corresponding assault rifle of each weapon tier.
• They carry a rocket launcher, which allows them once per mission to cause damage to multiple targets (on the highest level they can learn an ability to do it twice). Their rocket launchers almost always hit and always do the same amount of damage, so this is a very reliable ability.
• Most of their abilities further increase the amount of damage they can inflict.
• Their bullet storm skill (which is learned very early) allows them to fire twice per round when they don't move. Measured in pure damage potential, this ability is more effective than the rapid fire ability of the assault (which can't be used on two different targets and which has an aim penalty) and the double tap ability of the sniper (which can only be used every 2nd round). By the way: it also allows them to shoot once and then move out of the line-of-sight in the same round.

By the way: with the "Enemy Within" DLC, the assault class can theoretically out-damage the heavy. The "Close and Personal" skill grants them one free shot per round against an enemy within 4 tiles. This can be followed with "Rapid Fire" for a 2nd and 3rd shot in the same round. "Close Combat Specialist" can give them another free shot during the enemy turn when an enemy moves within 4 tiles for a total of 4 shots per round. But keep in mind that the assault can only use this potential when played in a very aggressive and risky way, while heavies can do their damage while camping behind a full cover.

A Heavy's primary role is damage output. Heavies have the earliest access to being able to fire twice in a round (Snipers need to be maxed rank to do that), making them extremely good early on. Their Rocket Launcher is also great early on because 6 damage is enough to kill just about anything until you run into Chrystalids and Mutons.

In my opinion, Heavies are wonderful in the early game, but underpowered later on. I found that their lack of aim (their base aim at maxed rank is 75, whereas Supports have 90) is workable in the early game because enemies don't have much to help them avoid shots. Your Heavies can reliably hit around 50-60% of the time, most of the time, which is decent. Later in the game, many enemies have special abilities that help them avoid hits. I found that, even if an enemy was wide open, my Heavies could usually only hit around 30-40% of the time, which is relatively unacceptable. Rocket Launchers are also a lot less exciting when they only take off 30% of an enemy's health. Their Heat Ammo is REALLY nice for taking out Sectopods, though. 30 damage crit ftw.

The best use for Heavies, if you can avoid getting too attached to your soldiers, is probably to take advantage of their being great early on to have them baby other soldiers until other soldiers are higher ranks, then sideline your Heavies for other classes. Personally, though, I get too attached to my soldiers, and can't side out my Heavies when they've been so good to me all game. :(