I'm a little more shaky on this but I can try, I haven't played WoW as much as I have played the prior games like Warcraft 3.
After Illidan and his allies were defeated by Arthas in northrend, Illidan felt the need to hide from the burning legion since he failed them and was likely going to be killed. He retreats to outland (the name for what remains of the planet draenor) along with his allies the blood elves and the naga.
Right before vanilla the alliance and horde in their current form are created without worgen, draenei, blood elves, and pandaren. The alliance and horde engage in many skirmishes and are at war with each other. The leader of the horde is Thrall, the orc who was once enslaved by humans who became a powerful shaman, who is also the orc racial leader.
Multiple threats to the world such as the elemental Ragnaros, the black dragon Onyxia, and the old god C'thun arise. They are defeated by groups of adventurers (the players). These are all the classic raids if you want to play through them/the storylines surrounding them at some point.
The dark portal, the portal between azeroth and draenor that the orcs originally came through in their invasion that was later destroyed, is somehow opened again allowing passage between azeroth and the remains of draenor that are now referred to as outland. The draenei flee from outland and crash land in azeroth on their ship called the exodar, which they make their capital city in-game. The night elves and a group of human sailers discover the draenei, and the draenei decide to become part of the alliance.
Some of the blood elves come back from outland and decide to try and rebuild their capital that was destroyed by Arthas earlier. They join the horde.
The burning legion attacks outland and the horde and alliance head out to fight them. The adventurers also fight illidan and his allies. One of illidan's allies betrays Illidan and joins the burning legion, and attempts to summon a burning legion leader kil'jaden into azeroth. The summoning is stopped by a group of adventurers and azeroth is saved.
In outland the orc leader thrall meets the son of his good friend who died prior to the events in wow. The son's name is Garrosh Hellscream and he joins the horde as a general.
After the events in outland a plague strikes azeroth, originating from northrend from the lich king. The horde and alliance both go to northrend to fight the lich king.
One of the most important battles in northrend was the battle at the wrathgate, a large gate that stands at the doorstep of the lich king's domain icecrown citadel. The horde and the alliance were fighting together against the lich king's forces (the scourge) and the lich king himself came out to fight them. At this point the forsaken appear and betray both sides, as they attack everyone there, both horde and alliance, with a chemical weapon (called the new plague) in an attempt to kill the lich king. The lich king survives but almost everyone else dies. The lich king manages to take one of the heroes of the alliance with him as a prisoner, his name is Bolvar Fordragon.
Eventually the lich king is defeated. Arthas is killed, but the armor that Nerzhul was imprisoned in is not yet destroyed. Bolvar Fordragon wears the armor and fuses with Nerzhul, becoming the new lich king as he believes for the good of the world there must always be a lich king.
Everyone leaves northrend and goes back home. The giant black dragon Deathwing awakens and azeroth is struck with a mighty earthquake and many areas are destroyed or altered by Deathwing. Thrall resigns as leader of the Horde, as by now he has become the worlds strongest shaman and feels it is his duty to help the world recover instead of being warchief. Garrosh Hellscream takes over as the new warchief of the horde.
The worgen are attacked by the forsaken and seek help from the alliance. They end up joining the alliance. The goblins also join the Horde at this point because of their own problems.
Deathwing is eventually defeated, though tensions between the horde and alliance are at an all-time high with the bloodthirsty Garrosh at the head of the horde. A new continent is discovered, Pandaria, and both the horde and alliance venture there.
Pandaren are the natives of pandaria, and some choose to join the horde and others the alliance. Many remain neutral. Lots of conflict happens between the Horde and Alliance, along with much internal conflict between races within the Horde. Garrosh Hellscream, warchief of the Horde, eventually goes rogue and betrays the rest of the Horde. He is defeated by a coalition of both the Horde and Alliance and imprisoned, to be put on trial. Vol'jin, the racial leader of the trolls, is now the new warchief of the Horde to replace Garrosh.
Garrosh escapes from prison and time travels to Draenor right before the orcs were corrupted by Gul'dan and the burning legion. This all happens in an alternate dimension. Garrosh then prepares the orcs in this alternate dimension draenor to invade azeroth in the main dimension, creating a new horde he calls the Iron Horde. Garrosh is defeated and killed by Thrall, and the Iron Horde is dismantled.
This is where we are currently. The next expansion has already been revealed by Blizzard, and is going to involve the Burning Legion invading Azeroth again. Illidan is also revealed to have not actually died when we fought him in Outland and will play a part in the fight against the legion.
There are some more things people have discovered by data mining information from the Legion alpha (or just participating in it) but that's all that has been officially revealed or covered thus far in Warcraft games.
This post is a lot less reliable than the previous one, even with the previous one having errors. I haven't played enough WoW or read enough to feel I did it justice here. THis should however give you a general idea of what has happened. Anything involving the alternate universe draenor is also especially complicated (to me at least) and there are a lot of complications I don't know the answer to.