What type of file is ~$file?

It's a temporary file that disappears when you close Word.

Description of how (and why) Word creates temporary files (in the same directory)

This particular file is actually an Owner File.

From that KB article:

Owner File (Same Directory as Source File):

When a previously saved file is opened for editing, for printing, or for review, Word creates a temporary file that has a .doc file name extension. This file name extension begins with a tilde (~) that is followed by a dollar sign ($) that is followed by the remainder of the original file name. This temporary file holds the logon name of person who opens the file. This temporary file is called the "owner file."

When you try to open a file that is available on a network and that is already opened by someone else, this file supplies the user name for the following error message: This file is already opened by user name. Would you like to make a copy of this file for your use?

Actually, it is a temporary "lock" for the file that you are writting.

If someone else try to edit the same file(if you are editing a file on a network share for example), this lock will disallow simultaneous editions. Here is a quick reference.