- DEM = digital elevation model
- DTM = digital terrain (topography?) model
- DSM = digital surface model
Sort of the same thing - distinctions can be made if considering just terrain or also objects on that terrain. See What is the difference between DEM, DSM and DTM?
As for Grid Format, as mkennedy says, my guess would be the native raster format Esri uses. You would have to speak with your client to confirm. To create one you can export your DEM tiff. When specifying an output location, just point to a folder. Whenever exporting a raster in Arc, if you do not specify a file extension the result is a GRID file. You may also be able to choose GRID in a 'Format' dropdown depending on what dialog you're looking at.
Note that this format isn't as self-contained as a tif would be. If you create a new, blank folder and select that as the output location, you'll get two subfolders - one with the filename you specified and an INFO folder. There may also be .xml and .ovr files outside those two folders. It is important that all of this be included when transmitted - particularly that INFO folder. It's a required part of the raster.