How do you miter the end of a baseboard that stops flat on a wall in order to add the missing baseboard to that other wall?

You place a piece of the baseboard on
top of the piece you are going to install.
With a pencil draw the outline of the baseboard and cut with a coping saw.

If you are working in a corner and with baseboard which has a profile (other than flat) what you would do would be to miter @ 45 degrees the board that will but up against the one which is existing then follow the line that the miter creates across the profile with a coping saw. This will create an exact match to the existing profile and eliminates the gap which can ccurr with a basic miter.

If you are working in a mid-wall location the best joint is known as a scarf joint. Basically miter both pieces @ 45 degrees vertically (perpendicular to the floor). Each piece should be a mirror image of the other. The object in both cases is to avoid a crack which can be looked into directly. I could draw you a picture better than I can describe.

Good Luck