You are talking about a couple of different things. Someone holds the rights to the song - maybe a record company, a licensing company, or even, sometimes the songwriter. So when that song is played someone is owed a license fee. If you put it on a cd and the cd gets played that wold also be an infringement. In a nutshell, the live performance (as opposed to the sale or distribution of cds ) of cover songs is governed by 3 main performance rights societies: ASCAP, BMI or Sesac. Live music venues are supposed to buy a license from each of the 3 societies each year. This allows songs to be played live at their clubs. The societies come up with an equation as to how they distribute the license fees they collect to the artists that are members. These societies also collect license fees from radio stations as well. So, when a cover band plays at a club, the rights owenrs are supposed to get payment from either bmi, ascap or Sesac.
The same goes for when songsare played at sporting events (when a player walks to the plate, when a player is sent off to the penalty box, etc.), the venue pays....same for when a marching band plays a song at a college football game. Paul McCartney owns the rights to dozens and dozens of college fight songs. Next time you go to see a band at a club, look for the stickers on the door that serves as proof that the venue is paying their license fees. Many clubs of course don't pay the fees and get away with it. It's the old "if a tree falls in the woods ..." issue. Small clubs will not pay the fee and hope tha they won't get caught and they usually don't get caught. You should go to ascap's website to find out more information and to consider joining as a member. The site link is below Good Luck
Some quick background: The "covered" song is protected by two copyrights -- a sound recording copyright that protects each particular performance of the song and a "musical composition" copyright that protects the song's underlying words and music. Sound recording copyrights are typically owned by the performer's record label while the musical composition copyright is usually owned by the songwriter or a music publishing company.
If you perform a song in private and record it on a CD you have no obligation to pay any performer who previously performed the song or the owner of the copyright in that particular performance. You do, however, have an obligation to pay a "mechanical" license fee to the owner of the copyright in the musical composition.
As Oscar very ably notes, if you perform someone else's song in public you ALSO have an obligation to pay a "performance" license fee to the owner of the copyright in the musical composition. You will be well served by following Oscar's advice on that subject.
You are, however, also pressing CD's with your covered songs and so you need to contact the Harry Fox Agency to acquire a mechanical license to fix each of the covered songs onto the CD (i.e., to reproduce the musical composition). Visit http://www.harryfox.com/public/licenseeServices.jsp and spend some time learning the basics of mechanical licensing. If you have any questions afterwards you should contact a copyright attorney. Have fun.