Usually used in the south in reference to a drink of whiskey or hard liquor. The amount can be a swig or a gulp, but usually decided by the person consuming. The term is only used when one party offers to another party as a form of sharing. It is believed the term has evolved from offering chewing tobacco as a gesture of kindness.
person 1: What ya got there?
person 2: Hell, this is some Jim Bean. Why don't you give yourself a lil snoot.