History: Archmage Khadgar was sent as a young boy to apprentice under the last guardian, Medivh. Khadgar eventually became suspicious of Medivh's motives. Eventually, he figured out that Medivh was trying to open a portal between Draenor and Azeroth. Khadgar and Anduin Lothar rushed to stop Medivh, only to find that he'd already opened the portal. Khadgar and Medivh fought, and Khadgar managed to kill his old master, although he was cursed with frailty and old age in the process. He later fought in the Second War against the Horde and took the opportunity to lead the Alliance Expedition to retrieve Medivh's book from the Outlands.
Turalyon was the second-in-command to Anduin Lothar during the Second War. He fought one-on-one with Chief Orgrim Doomhammer and was responsible for catching him after the battle of Blackrock Mountain and driving the orcs back to the Dark Portal. He was picked by King Terenas Menethill II to join the Expedition a few years later.
Alleria Windrunner was a ranger-captain of Quel'Thalas. While fighting the Horde in the Second War, she heard her sisters (Sylvanas and Vereesa) had died, which was not true ... yet. This drove her into a rage, and she created an army of rangers specifically to hunt down any free orc in Azeroth.
Kurdran Wildhammer was an Aerie Peak dwarf who went into battle with Sky'Ree the Gryphon. He killed nine dragons during the Second War. He went on the expedition in hopes of hunting down Deathwing.
Danath Trollbane led a human militia in Stromgarde. He was well-known for his heroism at the battle of Khaz Modan, and later oversaw an orc internment camp. He joined the Alliance Expedition to keep Nethergarde safe.
Together, the five emissaries went through the Dark Portal to retrieve the Book of Medivh from the orc warlock Ner'Zhul. They found the book, but Ner'Zhul had already learned enough magic to destroy Draenor from the inside. Khadgar chose to close the Dark Portal from the inside, trapping the expedition on Draenor but saving Azeroth. The five heroes vanished into a portal and were presumed dead. The giant statues in the Valley of Heroes in Stormwind commemorate their sacrifice ...
Where they are now: ... but of course, this is Warcraft, where no one dies if it's at all feasible for them to be alive. You've probably already seen Khadgar -- he allied with the Naaru and now makes his home in Shattrath City. Danath Trollbane and Kurdran Wildhammer started and led two Alliance outposts in Outlands, Honor Hold (Hellfire Peninsula) and Wildhammer Stronghold (Shadowmoon Valley) respectively. Alleria and Turalyon are missing, although Allerian Stronghold in Terokkar Forest is named after the night elf. They also had a kid together, which I didn't even know was possible. Arator was left in Nethergarde when they went to Outlands and has now returned as a grown-up to search for them. He looks exactly like a blood elf paladin, which is less about lore and more about Blizzard being lazy with their models. On Hordeside, you can do a quest in Ghostlands to deliver a pendant from Alleria to Sylvanas. Sylvanas acts angry, but then she sings a beautiful song to her sister.
Wait, there was no Horde expedition. How did the Horde get camps in Outland?: The orcs, who are generally considered the leaders of the Horde, came from Outland, so they already had a lot of camps left behind and allied clans willing to help them. Plus, have you seen what the Horde considers a stronghold? It's two sticks, a tarp, and a bunch of pointy objects to stick in the ground. Oh, and a couple of hammocks.
For more information: WoWWiki on the Alliance Expedition, WoWWiki on the Valley of Heroes, Sylvanas sings to Alleria.
Edited for correction: Alleria was a high elf, not a night elf.