I can't vouch for most of the country but it seems in Arizona and Southern California, mountain lion is the most commonly used name, while in the northwest cougar is most common.
Northwest (Washington and Oregon) - cougar
Rocky Mountain West and Southwest (California, Idaho, Colorado, Utah, Arizona, New Mexico, Texas) - mountain lion
Southeast (Florida) - panther
I have heard it said there are 40 different names used, but these are the only three I ever hear used in the United States (plus puma which more scientifically minded people use). Some other often cited names like "catamount" or "painter" I do not think have been used since the 1800's.