This post exists because of the massive spoilers in this link, but the post itself will be spoiler free. As long as you don't click on that spoiler-heavy link, you will not see any spoilers in this post. (Edit - actually, there's one spoiler at the very end of the post - it's clearly marked as such, and it is a minor spoiler at best, but it is there. Let that guide your actions.) Instead, we're going to talk about the position of Warchief - how it came to be, how it evolved and then devolved, and how Garrosh Hellscream's reign as Warchief set the stage for what could be a completely new direction for his successor (whose identity I will not discuss).
The position of Warchief actually began as a complete figurehead, and the first orc to hold that position, Blackhand the Destroyer, was placed in that position due to his combination of physical fearsomeness and egocentric self-aggrandizement - so easily was he misled and directed by Gul'dan, head of the Shadow Council and architect of the Horde, that he never once proved himself a threat sufficient for Gul'dan to ever consider replacing him. It's not that Blackhand was either a fool or an idiot, he was in fact a canny tactician and a respected warrior. He simply believed his own hype - so convinced was he in his own superiority that when Gul'dan presented to him that he would be a respected equal and his position as Warchief would be one of real power, he believed it, because he believed in himself. Throughout the war with the draenei and later, the invasion of Azeroth, Blackhand ruled as Warchief and allowed himself to listen to Gul'dan's words - allowed himself to listen because they were telling him what he wanted to hear.
Even as the humans balked the orcs, and Blackhand's series of victories became defeats, he continued to listen to Gul'dan. This would be his downfall.
Gul'dan was a master at that, however - this is an orc who created the death knight on the spur of the moment to save his own life - and so he hit upon a plan. He resurrected ancient orc traditions from the time of the orc/ogre wars, when the gronn pushed the ogres down from the mountains and into the fertile lowlands of Nagrand - back then, the orcs had been driven to the brink, and had only managed to preserve their people via unity and a common leader. A chief among tribal chiefs. A lord of the clans, if you would. Humans would call this kind of ruler a high king, but orcs have no kings. And so, Gul'dan offered to Blackhand the position of Warchief - he would be the first to hold it in untold generations, since the orcs were no longer threatened by the ogres. Now, the draenei threat would be dealt with.
It was to avoid this fate that Gul'dan accepted the offer of the human wizard Medivh, who was himself possessed by the archfiend Sargeras (Kil'jaeden's erstwhile master) - and in so doing, Gul'dan's puppet Warchief found himself fighting a new war against a new enemy on a new world. And it was because of that war that the position of Warchief went from that of a puppet to sinister warlocks to a real position of power, because Blackhand the Destroyer simply was never intended to win that war. Gul'dan cared nothing at all for his people, and absolutely didn't care if they won the war against the humans. He cared for finding the Tomb of Sargeras, and so, when humans discovered Medivh's treachery and moved to kill him, Gul'dan took the opportunity to attempt to ransack the dying wizard's mind. What he found there nearly killed him. It was in this moment that Orgrim Doomhammer seized control of the Horde, and in so doing elevated the position of Warchief from that of a figurehead to a position of absolute power. Doomhammer killed Blackhand, assassinated much of the Shadow Council, and ended their ability to command the Horde. Finding himself the head of a military organization waging a war they were losing, Doomhammer dedicated himself and his Horde to the conquest of the Azeroth they found themselves stranded on. The sum total of the end of the First War and the complete Second War, from the destruction of Stormwind to the raid on Lordaeron and the defeat of the Horde at Blackrock Spire is the story of the change between the Gul'dan led Horde and that of Doomhammer as Warchief, and the error Orgrim made was one of expediency - with his people stranded on an alien planet, he decided they needed Gul'dan and his warlock magic if they were to win. He made the mistake of thinking he frightened Gul'dan enough to be able to compel his obedience. But in his way, Gul'dan was brave - he sought godhood, and in his quest abandoned the Horde outside Lordaeron.
Still, the modern Horde's idea of what the Warchief is comes from Doomhammer, and Thrall became Warchief in direct succession from Doomhammer, as the aged orc fought to free his people from the human internment camps they'd languished in since the end of the Second War. When Thrall established his reign and led his people to Kalimdor, he did so with the unquestioned authority of an autocratic supreme ruler, even as he made friends and allies along the way. The tauren and trolls who joined the Horde were used to being able to bring their concerns to Thrall and speaking their minds freely, because Thrall was a reasonable and fair minded individual, but nothing in his position forced him to listen to them. He chose to do so. When he issued commands, he did so as the unquestioned leader of all the Horde, from the orcs to those other races that had come under its banner. Interestingly, even as Thrall allowed forsaken, blood elves and even goblins to join, he made no provisions for the Warchief being anything other than an orc ruling in an orcish manner.
As a result, what we had was the system of government invented by a madman whose evil lust for power led him to sell his people into demonic slavery as a trick to dupe those people, used by another orc who murdered his predecessors, inherited by an orc who never actually lived among the people he was using it to govern until adulthood and who chose to keep that autocratic system even as he invited many people who were not orcs into the organization so ruled. The position of Warchief was that of the chief of all the orc clans. Therefore, when Garrosh came to power, he chose to rule the Horde exactly as he understood the position of Warchief ruled - namely, as chief of all the orc clans. He treated the other races who'd come in under the Horde banner at best as adjuncts, as Blackhand had treated the ogres and goblins - even Doomhammer's temporary alliance with the Amani trolls under Zul'jin was beyond his understanding.
While you don't need to have played the previous Warcraft games to enjoy World of Warcraft, a little history goes a long way toward making the game a lot more fun. Dig into even more of the lore and history behind the World of Warcraft in WoW Insider's Guide to Warcraft Lore.