Half-Ogre
The Half-Ogre is, as its name suggests, the fruit of the inevitable union between an ogre and a member of some other species. The resultant creature is tall, strong and hardy, if not quite as brawny as its ogre progenitor, but also possess considerably more brains than its pureblooded ogre kin - admittedly, it may still seem somewhat dimwitted by human standards.
Dungeons & Dragons
In Dungeons & Dragons, the half-ogre's history is a rather unglamorous one. See, they've been around since the days of Advanced Dungeons & Dragons 1st edition - and none other than Gary Gygax himself wrote up an article on half-ogre PCs in Dragon Magazine #29, with Roger Moore posting a follow-up expansion article in Dragon #73. But, at the same time, the half-ogre's been pushed to the background, save for warranting a place in the Monster Manual.
The reason for this is two-fold. Firstly, the "big dumb ugly monster-blooded bruiser" niche is already amply filled by the half-orc. Secondly, given that an ogre is stronger than an orc, you'd expect a half-ogre to be likewise stronger than a half-orc - this makes half-ogres extremely hard to balance.
And yet... they've clung on for dear life. Even if the goliath may have shoved them out of the limelight forever since it got promoted, the half-ogre was a PC option in the first three editions of the game.
AD&D 1e
As mentioned, the 1e version appeared in two parts in Dragon issues #29 and #73.
AD&D 2e
The AD&D 2e Half-Ogre appeared in the Complete Book of Humanoids, alongside both its Ogre progenitor and its Oni cousin.
3e
There are actually three versions of the Half-Ogre in 3rd edition; one in Dragon #313, one in Races of Destiny, and the Kyrnnish Half-Ogre in the Dragonlance Campaign Setting, which was reprinted in Races of Ansalon.
Pathfinder
In Pathfinder, things are slightly different. There, ogres are disfigured horrors right out of a hillbilly slasher flick - distorted monsters mutated through generations of incest, Lamashtu worship, toxic food and who knows what other foulness. Of course, as a species of killer hillbillies, Pathfinder's ogres love them some rape. Most of the time, they don't remember to stop before their victim dies, but when they do - or when it's a horribly mutated ogre gal snu-snuing some unfortunate guy - the result is a horrible mutant. See, ogre genes are so foul and polluted that any creature with ogre blood in its veins winds up a disfigured abomination in their own right, and that's one gene-pool that won't ever come right. As a result, these ogre-blooded freaks tend to be just as nasty and inbred as their ogre progenitors - maybe more so.
In Pathfinder, though, these unfortunates are called Ogrekin; a "half-ogre" is what you get when an ogre mates with a giant, and whilst such critters are still deformed, they come off a lot better than the poor ogrekin do.
Ogrekin status is handled as a template that can afflict Medium sized humanoids. There's two separate versions, but the idea remains fairly consistent.
Ogrekin Template, mk 1
This is the first version of the ogrekin template, which appeared in the original printing of "The Hook Mountain Massacre".
Ogrekin Template, mk 2
This is the finalized version of the ogrekin template, which appeared in the Bestiary 2 and gained further mutations in the Inner Sea Monster Codex.