Do the bosses have particular weaknesses?

Solution 1:

Some notes on boss fights in the earlier difficulties, this does not address Hell and esp. Inferno with the enrage timers.

Butcher

Position yourself with a clear path to the healing wells during the battle. If you're not boxed in on the other side of the room, you can quickly retreat & refill hp if needed.
Don't stand in the fire. He is really big & a melee boss, if the tile you're on starts burning either move around him or lead him to a safer place.
Ranged characters, team up with a melee class or use Templar in single player. The follower will likely die often, but you can just run around for 30 sec till he rezzes.
Don't run away in a straight line; as he can use Ancient Spear to catch you.
He sometimes points a bit in the direction he will use Ancient Spear, which makes it easier to avoid.
Learn when to run away when the hammer is raised high and about to come down hard, there are other white mobs that have a similar tactic so you can practice on them instead of Butcher.

Belial

Move away when you see a green warning similar to meteor, that is his poison 'meteor' skill incoming
His poison breath attack sometimes doesn't hit at the extreme ends of the platform, so try to run there if you notice it in time.

Azmodan

Take out the demon summoning platforms asap to prevent any adds from joining the fight. In a team, this should be done by ranged while melee keeps him occupied.
When avoiding the black expanding circles, don't run too fast but retreat just outside of the circle. That way when he casts the next circle on you, it will overlap a lot with previous one and give you room to move around.
If he is off-screen, don't stand in the same spot while attacking - this makes it easier for him to charge you.

Diablo

Expect him to teleport near you when he is off-screen, and be prepared to run away.
In stage 1 & 3 in the main room, you have access to healing wells. In stage 2, you have no wells: So stock up on your highest potion and make sure its the active stack.
In the realm of terror, focus fire on a single target to take it out. The fog makes it harder to aim and see enemies quickly, so having an auto-aiming pet can help find them faster.

General strategy for team play

If an ally dies, try to lead boss away from the corpse so someone else can resurrect them. Similarly, if you're about to die don't do so in the middle of the room where the boss can keep interrupting someone in rez attempts.
The duration may be less, but the bosses can all be stunned etc. Using fast attacks which stun/snare/slow enemies will make the fight much easier.

Solution 2:

If there are weaknesses, they are not in the way for example pokemon works (beating fire with water). The elemental damage in D3 is different from the D2 elements. In a way elements have become a purely visual gimmick. Resistances are still important for your own equipment to survive certain champions but you can kill them either way. Information about the Boss HP is aviable but not about resistances afaik. If you want to exploit the bosses on the higher difficulties you should do it in the way of survivability vs dmg output vs movability. But you don't have to think about killing diablo with Ice spells instead of arcane.

Most bosses in hell and upwards can be exploited the same way they were in the lower difficulties: by evading their attacks efficiently to reduce their dmg output. Try to heal yourself up rather sooner than later so you don't get suprised when a sudden burst kills you.