Onyxia has a fair amount of abilities which come into play as the fight progresses.
Ony really only has four abilities to worry about in phase one. What they all basically boil down to is, if you are not the tank, do not stand in front of Onyxia. If you are in the raid at all, do not stand directly behind Onyxia. These are both bad places to be. In front of Onyxia, you will be subject to Wing Buffet, which will not only knock you back but will deal up to 24k damage (31k on 25's) to you (ignoring any armor) - this ability fires in a cone, so it's best to make sure you don't inch up into it by mistake. If this ability fires at the same time as her Cleave, any non-tank unlucky or stupid enough to be in front of her will be dead. So please, please, please don't stand in front of Onyxia if you are not the tank.
It's usual for the tank to run in ahead of everyone else and then move Onyxia into position facing away from the entrance to her cave so that he can ensure he won't be Wing Buffeted too far away from where he is tanking her, using the wall as a backstop.
She also breathes fire on people in front of her. You don't want that either. Fire bad. Fire very bad. Fire hit for 22k or so (as much as 30k in 25's) so all told, being in front of Onyxia is something only one very strongly motivated brick of meat should be attempting.
Meanwhile, while all of that horrible spiky and burny stuff is going on in front of Ony, the one place you may feel motivated to stand in (directly away from it all) may ultimately be worse for the raid. Onyxia also has an ability called Tail Sweep, and what that does is inflict a bunch of damage to you (I didn't take it in 25's, but in 10's it did about 6k to me) and toss you backwards. Now, to the left and right of Onyxia's cave as you enter are two side chambers full of whelp eggs. If you get hit by Tail Sweep, you have a good chance of breaking some or all of these eggs and spawning a whole lot of annoying little dragon whelp adds to come swarm your raid while they're trying to deal with that great big giant dragon, which is very very bad. So it's best not to stand directly behind her, as Tail Sweep is also a cone and can be avoided by standing to one side or another instead.
So there's phase one for you: don't stand in front or behind her unless you're the tank, who will be in front of her, and do that funky DPS. Once you get her to about 65%, the fun really starts happening.
Ony is a fickle lady, and she gets bored easily. At about 65% health she decides that she's not really interested in being stabbed, shot and set on fire (or whatever it is you're hitting her with) and she walks to the back of her cave and takes off into the air. That's when several things start to happen.
First off, she breaks a lot of the eggs in her whelp caves herself and unleashes about 40 whelps on the raid. This is fairly easily dealt with. The entire raid collapses to a center point and the tank (or tanks if you have more than one) gathers up the whelps, which are then DPS'd down by the raid. It shouldn't take long if the raid gives the tank enough time to lock down threat so they don't peel off of him and run around the room chasing that one nuke-happy mage who opened up before the tank had even reached them.
Yes, I'm looking right at you here. You know who you are.
After the whelps are dealt with, we still have another party favor, as Onyxia begins summoning lair guards. These are basically just beefier versions of the patrols you face cutting your way to Onyxia, with a couple of abilities to make tanking them interesting. First off, they cast Cleave just like their momma. Yet again the core lesson of raiding here is don't stand in front of anything. Secondly, they cast Ignite Weapon which gives them a huge AoE damage ability while it's up, so warriors or rogues should be using their various disarms to keep this mob from having a polearm equipped as much as possible. (I know hunters with Chimera Shot can technically disarm, but as ranged you're more valuable DPSing Ony, which we'll get to in a second.) Thirdly, they have a channeled Fire Nova that does a lot of damage to anyone in melee range if they don't die fast enough. Basically, since melee can't hit Onyxia very effectively during this phase (I couldn't at all) a good strategy for these Lair Guards is to have them tanked and burned by melee before they can do much damage, either killing them quickly or ranging the Fire Nova depending on how much melee DPS you have available.
If you have two tanks, it's best to have the off-tank tanking these and the MT picking up any that spawn while the current one is still up. I've seen reports that two are supposed to spawn at once every 30 seconds, but I didn't see more than one spawn every 30 in both 10 and 25, so pay attention as they may change this.
Meanwhile, Ony herself isn't idle. She's up in the air, pelting the raid with Fireballs that do significant damage (around 5k in 10 and 8k in 25's) and also breathing. Like humans, Onyxia takes a lot of deep breaths following the birth of her whelps. Unlike humans, her deep breaths create a patch of flames on the floor that ticks for about 9k damage every quarter of a second. This is a cone cast on the ground directly in front of where she's facing while she's airborne, and it can be avoided by not being where she's facing when she's casting (it's an 8 second cast, meaning that you should have plenty of time to get out of the way especially if you have someone calling it).
As long as people don't stand in the burning floor (and really, now, if you've played WoW at all the past four years you've already learned that the floor is where all evil lurks and you're properly suspicious of it) the ranged can simply keep pumping as much damage as possible (especially damage over time spells which continue to tick while everyone is running away from the cone of burning death on the floor) until Onyxia is at 40%.
At this point she lands. Finish off any remaining Lair Guards fast (this is why we have the MT trying not to be tanking any adds if possible) as the tank picks Ony up. Get some Fear Wards out, drop some tremor totems, do anything you can think of to break or interrupt fears because here comes phase three, the original 'running around like a chicken' phase.
Phase three basically repeats phase one with two exceptions. You still don't want to be standing in front of or behind her, as all those Cleaves, Fire Breaths and Wing Buffets still apply. (Again, if you are the tank, go ahead and stand in all that.) The two new abilities added to the mix are Bellowing Roar - everyone runs around screaming "OMG I just remembered I'm fighting a huge freaking dragon we're all gonna die aaaaaaaah" for three seconds unless various fear breaks are used - everything from Will of the Forsaken to Every Man for Himself to Berserker Rage to Fear Ward or Tremor Totem - heck, a PvP trinket works if that's all you have access to - and Eruption, where molten hot magma bursts forth from the incredibly visible cracks in the floor.
Ah, the floor. Source of all that's deadly and corrupt in the World of Warcraft. Horde and Alliance could be at peace if not for the floor. In all sincerety, the only time you should take any damage from Eruption is if you're unlucky enough to be feared into it.
There are also a few straggler whelps to deal with from time to time, but they're almost more atmosphere than an actual threat at this point. Just burn her down and collect your shiny new retro skinned loots, I say. Congratulations! You've just killed Onyxia. Again. You monsters! She had a family! And you killed all of them, too!
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