LUT stands for lookup table. A 3D LUT is a cube-shaped grid laid over every possible colour: red along one edge, green along another, blue along the third. At each point of the grid, the file stores the colour that input should become. A 33-point LUT has 33 × 33 × 33 = 35,937 entries; a 65-point LUT has 274,625. That sounds like a lot, but a 10-bit video can contain over a billion colours, so nearly every pixel falls between grid points and its new colour has to be worked out from the neighbours.
Tetrahedral versus trilinear
Trilinear interpolation blends the eight corners of the little cube a colour falls in. Tetrahedral interpolation first splits that cube into six tetrahedra, finds the one containing the colour and blends just its four corners. The tetrahedral split keeps the diagonal from black to white exact, so greys stay grey, and it avoids the slight hue shifts trilinear blending can add around strong colours. It costs almost nothing extra on a graphics chip, which is why it is the better default.
1D LUTs and .3dl files
A 1D LUT is three separate curves, one per channel. It can change contrast, gamma or overall tint but cannot move one hue without moving others, which is why looks are almost always 3D. The .cube format is plain text with a size header and rows of decimal numbers between 0 and 1. The .3dl format stores whole numbers instead and lists its grid positions on the first line. Both open here, so use whichever your source provides.