When you sign your name in your own handwriting, that is your signature. Things you can put your signature on include checks, permission slips, letters, stays of execution, and declarations of independence.
Signature isn't just about signing your name on the dotted line. A characteristic mark, tune, or style is also called a signature. Perhaps you have a favorite outfit or style that you consider your signature look. "I Did It My Way" was a signature tune of Frank Sinatra. In music, the signature indicates what key the song is set in by showing the sharps and flats.