Well, they're "lookup tables". Does that help?
If your answer to that question was a resounding, irritated, and violent "no", then you might want to check out this helpful tutorial, in which Casey Faris breaks down what LUTS are, how to install and then apply them in your projects -- perfect for those who have absolutely no idea what the hell they are or what they're used for.
But before we get to that video, here's Ground Control to explain the absolute basics of what LUTs are -- with cats:
And now for the nitty gritty from Casey Faris:
Now that you know what a LUT is, it might be helpful to know what it isn't -- or what it shouldn't be. It isn't (or shouldn't be) a glorified color correction filter. LUTs are used to change the color of an image, which, yes, means that they assist in color correction, as well as calibrating monitors, but don't think for one second that applying a strikingly beautiful LUT to your footage will make your footage strikingly beautiful. In fact, if your shots within a single scene differ in any way, whether in contrast, color, brightness, etc., your LUT may only make these variations more apparent. Be sure to have some uniformity in your footage first, because even though LUTs are powerful color modifiers, they aren't that powerful.
To get started, download some LUTS and make sure you have a LUT-loading tool -- a piece of software that will know what a LUT is and can load them and apply them to your footage. So, if you have Premiere, FCP, or pretty much any color-grading application, you're good to go. (What other programs are you going to be editing video in anyway, right?)