Refers to a man or woman who gets overly attached, obsessed, or clingy in relationships. Can also refer to an unwanted person who attaches to people of the opposite sex in social situtations (i.e. at a bar/club).
From the word barnacle: crustaceans which attach permanently to a hard substrate.
"Dude I have to break up with this chick, she will not leave me alone for one second. She is a total barney"
"Sarah, that barney from last week just walked in. Lets get out of here or we'll be stuck talking to him all night."