The man who introduced the world to the concept of "wholesale" destruction, Cohen is the Discworld's greatest warrior hero, renowned across the Disc for his exploits rescuing maidens, destroying the mad high priests of dark cults, looting ancient ruins, and so on.
On his first appearance in the series he is already an old man, but still tough enough to handle anything the world can throw at him; his opponents often underestimate him because of his age, realising too late that a man who does for a living what Cohen does and nevertheless survives to such an age must be very good at it indeed. Cohen does not know how old he is exactly, but estimates that he is between 90 and 95 years of age.
Cohen is described as a skinny old man, with a long white beard hanging down below his loincloth, and wearing a patch over one eye. His most distinguishing feature, however, is his smile — his unique dentures are made out of troll teeth, which consist of pure diamond and were inspired when Twoflower showed him his own (more typical) set.
The greatest problems now facing Cohen come from outliving the heroic age and finding himself in a civilised modern world where great battles and astonishing rescues happen rarely except in stories — which is ironic given that the Discworld runs on narrative.
One of the rare Discworld short stories, "Troll Bridge", tells of Cohen setting out to slay a troll, only to end up reminiscing with it about the good old days when things were black and white and everyone respected the traditions. Part of Cohen's danger to normal people is that as a barbarian hero he has extreme problems interpreting such things as empty bravado — as a man of his word, he naturally assumes that anyone else saying something like 'I would rather die than betray the Emperor' fully means it. This led to the deaths of several guards and courtiers in the Agatean Empire before everyone wised up.
In Interesting Times Cohen became Emperor of the Agatean Empire, having conquered it with his allies, the Silver Horde (see below). This is possibly a reference to Conan becoming King of Aquilonia, as well as Genghis Khan's Golden Horde and the conquest of China (it was in this book that we learnt Cohen's first name). This was intended to be a sort of retirement plan, but Cohen and his chums became bored and then abandoned the Empire in The Last Hero, in which Cohen decides to express his displeasure with the modern world by "returning fire to the gods, with interest". After the rather unsuccessful attempt, he and his friends escaped on the backs of horses belonging to the Valkyries and rode into the sky, seeking to explore the outside of space. His current whereabouts are unknown.
Given a barbarian hero's attractiveness to nubile young maidens, Cohen has quite a lot of children; in The Last Hero he mentions casually that he has dozens. The only one mentioned by name in the novels is Conina, who appears in Sourcery. She wishes to be a hairdresser, but Discworld-style genetics keep getting in the way, causing her to instinctively kill people who threaten her. She was last seen in an amorous relationship with Nijel the Destroyer. Interestingly, she says she actually knew Cohen and that he took an interest in her education — such as setting a length of corridor with a variety of traps for some heroic training.
In Interesting Times, The Silver Horde aid Cohen in his invasion of the Agatean Empire in an effort to steal something, which is hinted at but not revealed until the end to be the Empire itself. They also have a hand in overthrowing the current Emperor (a cruel tyrant who isn't "simply at Death's door but well inside the hallway, admiring the carpet and commenting on the hatstand"). A main point of the plot is Teach's attempt to civilise the Horde, a difficult task since "every one of them saw a book as either a lavatorial accessory or a set of portable firelighters and thought that hygiene was a greeting."
With Cohen crowned Emperor, the Horde live like royalty, all except Teach, who dies in the final battle of the novel after proving his barbarian credentials beyond all doubt with a berserker-like rage that amazed even his cohorts.
As of The Last Hero, Old Vincent is also dead, having choked on a cucumber (or possibly a concubine — there's some confusion on this point in the dialogue). The Horde sets off to return fire to the gods in a glorious last adventure, but eventually realize that this would destroy the Disc and, it is implied (if not explicitly stated) that they give their lives to save everyone else.
It is unsure if the Horde actually die in rescuing the world from destruction. They hurl themselves and a massive explosive device off of a frozen mountain cliff, and once the blast has cleared, Valkyries arrive with the intent of collecting fallen heroes and taking them to a glorious afterlife of drinking and fighting. However, the Horde suddenly rises up from the snow and steals the Valkyries' flying horses, deciding that they were bored with the Discworld and that it is time to explore other worlds. When asked whether or not they are truly dead, Cohen stated that they didn't think they were dead, so why should they care what anyone else thought? They never had before.