Here's an excellent video on how to trim a window. The presenter installs, however, a wooden stool. Here's a crappy video which shows installation of a marble stool. Basically, the installer attaches a marble apron below the window and then uses the apron to support the marble stool above it. -- amateur

"Stool" is correct for the interior but if I say "sill" to a client I don't have to explain it, even though a sill is exterior piece that is typically sloped, under older windows.

The way that window is installed, being flush or nearly flush with the wall surface is not really conducive to having a stool. Older windows commonly had a 1 1/4 (or so) set back toward the exterior, and the rabbeted stool attached to the sill. Newer windows often have a return as well ("inset") from the interior view, as the nail fin keeps them from coming in flush. Either situation lends itself fine to a stool as there is a nice ledge for it to sit on.

With a flush set window like this you'll be "faking" a stool and you won't want to make it very deep to avoid projecting too far into the room. Much easier to do with wood. If you use marble I suppose that using something thicker as an apron for that marble to sit on could facilitate that, although whatever construction adhesive is used to adhere it won't likely hold forever.

A stool or apron is basically cosmetic. I like a picture frame look with no stool.