In the first two Warcraft games, the Horde is made of the orcs under the command of the Burning Legion and are enemies of the human led Alliance. The orcs attempt in both games to conquer the human kingdoms. Eventually the Horde was defeated, most of its leaders killed, and the orcs placed in internment camps.
In Warcraft III: Reign of Chaos, the Horde is led by the young warchief Thrall after he led an uprising to free the orcs from the vengeful humans of Lordaeron. Thrall then led the orcs across the sea to Kalimdor after having a dream with a prophet who told him to travel across the sea to find his destiny, allying with the Darkspear trolls and the tauren. By the time of the orc campaign in Warcraft III: The Frozen Throne, the Horde also includes a few goblins, and it was revealed that the orcs first came from Outland, the shattered remains of their original homeworld Draenor. Later on in the reign of chaos campaign, orc commander Grom Hellscream and his men drink the blood of the pit lord Mannoroth and became corrupt themselves once more, eventually leading to Grom's death.
By the timeframe of World of Warcraft, the Forsaken, a faction of undead who rebelled against the Scourge, have also joined the Horde. The blood elves join the Horde early in the storyline of World of Warcraft: The Burning Crusade. Early in the storyline of World of Warcraft: Cataclysm, Thrall steps down to deal with the Cataclysm, appointing Garrosh Hellscream, son of the former traitor then redeemed hero Grom Hellscream, as warchief, and the goblins of the Bilgewater Cartel join the Horde. World of Warcraft: Mists of Pandaria saw the Huojin pandaren join the Horde.
The Horde is led by a warchief (i.e., warlord of the Horde). The Horde respects combat prowess and decisive action. Sometimes the position of warchief is earned through war and domination; sometimes it is won through duels. Other times its successor is appointed by the previous warchief. They are now led by the troll Vol'jin, since the ending of 'Mists of Pandaria'.