Nothing really...
Messerschmitt was the designer...and Bf stands for Bayerische Flugzeugwerke.
In 1926, Bayerische Flugzeugwerke (Bavarian Aircraft Works or BFW) formed. Messerschmitt Flugzeugbau also established around this same time. Both were separate companies.
In 1927, the Bavarian government forced a merger between the two.
In 1938, BFW changes its name to Messerschmitt AG. Aircraft developed after 1938 use the "Me" prefix, and since the 109 was developed in 1935, it carries the 'Bf' while the 262, 163, 323, etc., designed after 1938, carry the 'Me' designation.
Just a little piece of trivia :-)
Jeff