The Witcher games are based on the novels and short-stories of Polish author Andrzej Sapkowski, most of which were unavailable in English when the first game came out, but most of which have now been translated.
They follow Geralt of Rivia, a professional monster-slayer in a land with no straight-up good or evil, but rather shades of grey, where the greatest monster can have a point, and even the most amiable king can be ruthless when crossed.
Outside of the palaces and corridors of power, it's a violent, racist land, with monsters spawned from traditional fairytale mythology, and a perpetual struggle going on between the humans and oppressed non-human minorities. As a mutant, Geralt often finds himself caught in the middle, liked by neither, but better to have as a friend than an enemy.
The games take place after the books, which ended with Geralt's death. The Witcher had him mysteriously brought back to life without his memory, allowing for new stories to be told, but with a number of elements weaving their way back in, including his friendship with the bard Dandelion, who chronicles his stories like a randier Dr. Watson, and his love for the sorceress Yennefer, who he remains on the search for.
The word 'witcher' is an invented one for the games. The actual term used in the books is 'Wiedźmin', which has previously been translated as both 'warlock' and 'hexer' - the term used in a licensed movie and follow-up TV series that came out in 2001 and 2003.
Aside from being bred and trained to be a monster hunter, Geralt has many advantages over the humans he ends up fighting. He has access to magical Signs that help him both in battle and in dealing with the many people who just want to use him, as well as a body that can resist the toxic effects of potions that would kill a normal person. He's also sterile and impervious to disease, which doesn't go unnoticed by the women of Temeria he happily sleeps with as a diversion, even if his heart is firmly locked on the missing Yennefer. At the same time, his cat-like eyes make it impossible to hide his nature, even if he isn't carrying a sword, and the witchers' strength makes them every bit as feared as they are needed. He's also something of a relic. Monsters still haunt the countryside, but the world has moved on, with only a few witchers remaining to hunt them, and no more currently being trained. They're largely alone in the world, with few real friends, and their welcomes invariably end when the monsters are dead.
Geralt's adventures take place in Temeria, one of the most powerful of the Northern Kingdoms. It's a fiercely political continent, still sore from previous battles, and with particular fear aimed at the Nilfgaardian Empire, also known as The Black Ones. Inside its borders, there are constant clashes between the human population and the oppressed non-humans, who often band together as the freedom-hunting/terrorist organisation known as the Scoia'tael. As is typical for The Witcher's universe, neither is the 'good' side, and in most clashes, both will have both a point to make, and have committed some distasteful act that makes it difficult to have much sympathy. Geralt's only loyalty is to himself, although his strength and knowledge often means that his presence can tip the balance of power. It's up to you which way you tip it, and why. Just remember, a good deed seldom goes unpunished.
Geralt will soon discover this again at the start of The Witcher 2, where he's gone from poor witcher travelling the kingdom in search for answers to the personal bodyguard of the Temerian king Foltest, partly in payment for saving the monarch's life at the end of the first game. He has friends at the highest levels of power, a beautiful sorceress lover who understands him and he can trust... even if they're both aware that she's not Yennefer. Riches, respect, and more await. What could possibly go wrong?