The real song is actually protected by copyright. That means that it cannot be used without permission for commercial uses. This includes in movies, tv shows, and commercials. Don't worry though. You won't get sued for singing it at home for your own birthday party.
The original tune for the song was originally for a song called "Good Morning to All". Since then, it has been recorded with the words we now know, and as of 1934 it has been copyright protected. (See the history of the birthday song on Snopes.
The copyright isn't set to expire until at least 2030. That's nearly 100 years after the original copyright was secured!
"Happy Birthday to You" brings in roughly $2 million in royalties every year. Much of that has gone to charities.