LUT
1D LUT
The simplest LUT. A 1D LUT simply maps an input value to an output value. It treats all channels equally, so 1 input value has 1 output value (hence 1D). A simple 1D lut for gamma conversion from linear gamma to a gamma of 2.2 could look like the table above. This 1D LUT has 11 points, which is not very accurate. Usually a 1D LUT has 1024 entries, or a resolution of 1024. An important aspect of the 1D LUT is that it is reversible, it can be used both ways because you can just swap the two columns.
3D LUT
1D LUTs are too simple to convert between color spaces with different primaries or white points, which is why 3D LUTs were created. A 3D LUT is still a table, but it has separate values for R, G and B.
Note that 3D LUTs are not reversible. If you need to invert a 3D LUT you will need to find the inverse LUT file. It is possible to reverse engineer a 3D LUT to get an approximate inverse, but this is not very accurate and should only be used when you have no other option.
LUT usage
Instead of applying a LUT file your software might have a color space conversion tool or plugin that allows you to select the proper in and output color spaces. As long as the color spaces you need are available feel free to use those in stead. What is very important about color space conversions is that they are essentially lossless as long as you are performing them in a floating point bit depth. You can go from ACEScg to sRGB and back without loss of data in 32bit float, but be very careful when you render images to anything other than floating point EXRs! Floating point allows for negative values when you're converting to a smaller gamut, but even 12bit dpx files, which can hold a significant dynamic range, will clip those negative values.
Sometimes you run into a LUT that doesn’t specify both the input and the output colorspace. Remember that an image is always in a color space. So there's no such thing as a LUT with one color space. For example, sometimes people will talk about a rec709 LUT, or a P3 LUT. This is wrong and confusing. A LUT always converts from one color space to another. If a LUT does not mention two color spaces it often means the ‘from’ color space is implicit and it will most likely be a standard color space like sRGB. ICC profiles tend to be like this (be careful Adobe people!). Just be aware of this and test it out if you need to use a LUT like this.