Yeah, I mentioned a couple pages ago that this show could be especially difficult to follow if you're not already familiar with the story. I consider myself pretty familiar with the series, but even I was confused by how the episodes can switch 10 years, 20 years, 30 years, 40 years, or more back and forth without any explanation. If you watch the whole show, you'll get an idea of how the timelines fit together, but it's very jarring and confusing at first, even for people who have played all the games and read some or all of the novels.
As for Witchers, as Garrett said, they are all monster hunters. So basically around 1,500 years before the story begins, the world merged with another world (basically, the multiverse theory of there being infinite alternate universes is true in The Witcher ), which is called The Conjuction of the Spheres. This brought monsters to the world and those monsters were so powerful that average people couldn't deal with them like they could, say, a bear or a lion. A group of warrior-monks decided to deal with them and would use potions and mutagens to strengthen themselves. They usually conscripted young boys and put them through intense training, but only 3 out of 10 would survive the crazy mutagenic potions they were subjected to in order to become a Witcher. As a result, they would become extremely physically powerful, increased speed and agility, have longer lives than humans, heal much faster than normal, enhanced senses (such as being able to see in the dark), and be able to use basic magic (known as signs), among other things. But they would also become incapable of having children and look... rather creepy. Over time, the Witchers split into different schools and diminished in ranks, and public opinion got worse and worse, with false rumors about them having no emotions and being not too different from the monsters themselves. But even though people now fear and despise Witchers, they need them to take care of the monsters, so they begrudgingly pay them for their duties (originally the Witchers were almost like knights and didn't sell their services). At the time of the novels/TV show, there are very few Witchers still alive, Geralt being one of the last (and there are even fewer during the games, which take place over a decade after the main story of the novels). He also happens to be the only survivor of an experimental concoction of mutagens, which gives him an advantage over most other Witchers.
That should mostly cover the basics just about what Witchers are. In Polish, the word is "wiedźmin," which the author came up with as the male version of the Polish word for witch "wiedźma," and it was translated as "hexer" before the video games handled it as "witcher" (if I remember correctly).