After-the-dot-in-a-file-name, it is called extension.
The "base name," "basename," "primary name," "filename," "file name," or the "file."
I've always called the everything before the dot and the extension the "file name".
It's called the basename. In fact, there's a unix/linux command for it:
basename - strip directory and suffix from filenames
Drive:/File/Path/FileName.FileExtension
After the period is the extension, the immediate name before it is the file name, the path before all that is the File Path, and of course at the end (or beginning) is the drive.
BaseName is another name for FileName.
I'm not a Ruby or Linux guy, so I guess I missed the BaseName thing. It makes for all sorts of interesting naming convention hilarity. I'm in the
Filename.Ext
camp, although that, too can be a FileName. (or maybe a FileNameWithExtension).
Once upon a time, the term Leafname was used, although that typically included the extension. I mention this for historical value, and since its dropped out of usage, it might be redeemable for this purpose.