When describing my tessellations, I often indicate the type of grid and its density.

Grids are prefolded onto the paper to allow for easier creation/collapsing of repeating patterns. Grids are either square or equilateral triangles (60°).

Square grids are simple to describe because the paper usually starts off as a square. The square is then divided by halves until the desired grid is created (leading to 16-, 32-, or 64-cell grids) or divided into thirds, and then halves (leading to 24- or 48-cell grids.) On occasion, a rectangular sheet might be used, so the density for both width and height are noted.

Triangle grids (which allow for a mixture of triangle and hexagon patterns) can also be folded onto a square, and are divided vertically either by a doubling of 2s or a doubling of 3s, leading to the same vertical divisions as the square grid. 60° angles are folded to match the density. When I describe a grid as 32-triangle, it means there are 32 triangles stacked vertically on the square sheet.

Triangle grids can also be folded onto a hexagon, which allows for easier sectioning of the grid as it follows the same pattern of halving (or thirds->halves) as square grids do. The density refers to the number of triangles from one edge of the grid to its opposite edge.

There are common twists used in square and triangle/hexagon patterns, which typically describe the final shape as well as the construction method, that I use. You can find a detailed list at twistdatabase.com.

There are also some folds and collapses which don't involve twists, and are typically described by their methods such as a "120° pleat intersection". Most of these names I pulled from Eric Gjerde's Origami Tessellations book.