It's less important to go over the history of the orcs in terms of the wars of Azeroth. We've done it, many times. What's interesting to discuss is the orcish acclimation to Azeroth, and furthermore, Azeroth's acclimation to the orcs.
The orcs have changed during their time on Azeroth from a nation of blood-drunk servants of evil to a people leading a faction that seeks global dominance in the name of a legacy they've invented for themselves. Orcs today have a warrior culture that comprises elements from Blackhand's Horde, their past on Draenor, and a great deal derived from Thrall's efforts to create unity and give his people a culture again. While the modern orcish nation is led by Garrosh Hellscream, a brown Mag'har orc, it cannot be said that most orcs of the Horde really understand Draenor. The Second War ended more than 20 years ago, and many of the orcs of today are the children of those who fought in it.
This must be understood: Many orcs alive today on Azeroth have never even seen Draenor. Those who did last saw it 20 years ago. Azeroth is their home as far as they're concerned, either the only home they've ever known or the one they've known for decades. From the perspective of most orcs, Draenor is effectively gone. Oh, many of them are aware that Outland exists, and there are those orcs who have been there in recent years, but most orcs living today have never seen it at worst and saw it decades ago at best. To them, Draenor is nearly a myth, and Garrosh Hellscream becomes a mythic figure as a actual brown orc, an uncorrupted Mag'har who lived most of his life on that long-lost homeworld. It is this, as much as his lineage as the son of Hellscream, that has made him a legend among the orcs of the Horde today.
What they had been
At the end of the Second War, the orcs of the Horde were beaten, demoralized. It was barely conceivable to them that the humans could possibly have defeated them. They'd destroyed the draenei (as far as they knew, anyway) and, in the process of embracing the demonic magics of the Burning Legion, rendered their home more and more unlivable. They'd followed Gul'dan and Blackhand the Destroyer to Azeroth because there was really no other choice.
The Horde was an engine of conquest without a target, turning upon itself in a world that couldn't sustain them any longer. Azeroth promised a new world, one fertile enough to support them. Even when Blackhand died at Doomhammer's hands, he kept the war going because there was still no choice. They couldn't go back to a world stripped of its ability to sustain them. Doomhammer couldn't well maintain his people as a unified nation if he failed to win the war he inherited as Warchief.
This must be emphasized. It does not excuse the orcish people or any individual orcs of the time from either responsibility or culpability, but it still must be stated that Doomhammer, as Warchief, had inherited leadership of an army that had burned its own boats. Going back through the Dark Portal was an untenable idea.
So Doomhammer fought and led his people to the very walls of the greatest city the humans had built. To him, standing before the walls of Lordaeron, it must have seemed as though the destruction of Shattrath was to be repeated. With Gul'dan's betrayal and the Horde's defeats at Lordaeron and Blackrock Mountain, the orcs were forced to experience the first true defeat since Ner'zhul first convinced them the draenei were plotting against them.
What they became
The orcs who survived were imprisoned in internment camps rather than destroyed. They endured this fate for years and years until Thrall, the orc foundling discovered by Aedelas Blackmoore, freed himself, discovered his father's people, and became the first new orc shaman since Ner'zhul's folly caused the orc ancestor spirits and elementals to turn their backs on the orcs. This moment is doubly amazing because not only does it signify the return of shamanism to the orcish people, it also signifies that Azeroth had accepted the orcs. Despite their years of servitude to the Burning Legion, despite the Blood Curse that bound them to evil, Azeroth's elementals and spirits would from this moment on speak to the orcs. Their lost connection to Draenor's ebb and flow, their ability to understand the message that everything that is is alive, was replaced.
Following Thrall's rise to the position of Warchief (and imagine the irony that while Ner'zhul had hungered for the power and respect of direct command, Gul'dan had installed a pupper ruler rather than directly ruling himself, meaning that Thrall was the first and only shaman to rule directly as Warchief), the second great event occured to change the orcish character and their relation to the world. Grom Hellscream, the first orc to drink the blood of Mannoroth, would die striking a fatal blow to the pit lord and, in so doing, free his people.
These two events combined to forever redefine the orcs as a people. They did not erase what had happened. The Horde of today keeps the memory of Warchiefs past like Doomhammer alive. It preserves many of the traditions of the Horde of Blackhand. Those traditions, however, were taken from the distant past of the orcs and so were ingrained, impossible to remove. Thrall did not try.
Rather than remove them, he sought to change how they were observed. This has led to modern orc culture, which is not a return to their old ways. It is in fact a constructed hybrid of the Old Horde and the shamanistic culture that preceded it. It is not tribal, although members of tribes exist within it. It is not nomadic; it establishes settlements and fortifies them in black iron. It is rigidly hierarchical, rather than assembling in kosh'arg festivals to determine if it will take action as a unit. It rather assumes that it will be doing so.
The modern Horde is effectively welded together onto this orcish social structure that takes the Horde that served the Burning Legion and achieves syncretism with its old, pre-Horde beliefs. In essence, they've replaced demonic magic with their older, spiritual magics but held onto the rigid militarism that has defined them as a people since Gul'dan's Horde.
Making the future
This can be clearly seen at the Dranoshar Barricade, where partisans for Garrosh and Thrall stand side by side bickering with one another over the course the Horde should take. Thrall, by accepting the title of Warchief previously held by Doomhammer, made himself the conduit through which the military nature of the orcish people would be maintained, while almost eradicating the demonic. Those few warlocks that remain do so both because the modern Horde requires an understanding of the forces that once nearly destroyed them, and because it is easier to watch them from where they are. Yet Thrall failed to ultimately change his people.
Decades of conditioning under Gul'dan had altered them forever. Garrosh Hellscream inherited leadership of an orcish people who have no idea what orcs were like, orcs who believe his brown skin makes him a truer orc than they are. His response, to dedicate himself to orcs first and foremost, has the effect of hallowing anything they do in his name. If an uncorrupted orc says to fight, how can it be wrong?
By embracing the martial tendencies that were always there but accentuated by Gul'dan and the Blood Curse, Garrosh has unwittingly stepped fully into the role of Warchief as practiced by Doomhammer. It can be no other way, for the orcs have had decades to make those tendencies second nature, and those tendencies are an outgrowth of their natural evolution on Draenor in the first place.
Who they have become
Orcs have shown that they can grow and change over the years. Their adoption of first demonic and later arcane magics has shown that they can adopt new techniques and make them part of their own culture. And it is this flexibility that has made them who they are, really, a people who have adopted a kind of spirituality over a martial nature.
Orcs today are a mix of contradictions. Free of the demonic taint, they yet live in the same social structures that Gul'dan discovered and accentuated in their original society. They revere a past they don't actually understand. They embrace a shamanism most of them do not actually practice, instead favoring a militarism antithetical to its teachings of balance. They are a tribal people that do not live in tribes, a nomadic people that do not walk the land. Their reaction to the Cataclysm Deathwing caused is the same as their reaction to the corruption of the land Gul'dan caused, because in essence Gul'dan created in the old Horde a meta-tribal structure that replaced the tribes themselves. The Horde becomes the tribe itself and as such has endured past the point where Gul'dan and his dark patrons were relevant to it.
The orcs today are a people who are struggling to find their way through these contrary aspects of their nature. They will confront the future with them. It's time that they examine who they actually have been and who they have become.
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.