My experience is in Lightroom, not Darktable, but LUTs just take an RGB value at each pixel and map it to a different RGB value. The LUT (look up table) is just a big table containing these mappings. They are most commonly used near the end of the development pipeline to do color grading. It's a common way to do something like emulate the color response of a specific type of film, even if you shot digitally.

So, typically I would do my developing (lens profile, crop, exposure, etc.) first, then apply a LUT. After the LUT, then I would tweak the color sliders if there are any I still want to adjust. The color is usually the last step of the process for me.

All that being said, these days I don't really use LUTs any longer. I am mostly doing product photography using a flash in a controlled studio environment. So, I pretty much have some develop presets all dialed in.

My expertise is with Video, not stills photography so maybe take this with a pinch of salt but since there aren't many responses yet it might be of value. There's different types of LUTs but I think odds are you're probably thinking of "creative LUTs" or "Look LUTs" so I'm speaking with reference to those.

With video footage shot log, one typically does a transform process first to bring the footage in to colour space appropriate for the project by using a transform function specific to the camera and brand. This is because the LUT will have been designed expecting a specific starting gamma curve and colour space and typically needs to be transformed in to those first in order to avoid unexpected results.

While RAW is a bit different from Log, there are much the same principles at play. It starts out with no gamma-encoding or display colour space but when you open it in editing software, the program applies a debayer and a default color/tone interpretation. In that sense, the “transform” stage happens in the RAW development settings first and any LUTs would necessarily be happening afterwards.

Another good reason to keep a LUT at the end of whatever processing chain you have for your image is because, the way the values are remapped by the LUT means that depending on what the LUT does, you could end up with destructive results were you get clipping or oversaturation and it's best to make sure that that happens last so you can compensate for it ahead of time before you have lost information from this destructive process.

I've never used LUTs in darktable, but my understanding is that they're applied in a module, which means that they are incuded in your regular image editing workflow within darktable

If you want to create a LUT in darktable, there's the color look up table module (it also has a few presets).

For "external" LUTs (the kind you download) there's the LUT 3D module (I've never used this one, but I'm fairly sure that's what it does).