Unless you see otherwise, always assume a piece of music is copyrighted (that’s what the law says). Anything created since 1923 (and possibly earlier, if it didn’t originate in the U.S.) is under some form of copyright or another.
To legally put it on your webpage, you’ll have to pay a fee to ASCAP (at http://www.ascap.com). The web is considered the same as broadcast, and comes under that sort of regulation.
You could also ask permission from the copyright holder.