If you are performing and/or recording a cover song on a CD, a mechanical license is required, per copyright law.
A mechanical license is an agreement that allows users to record and distribute a composition they don't control. It is quite easy to obtain these licenses. This type of service will allow you to secure a mechanical license and will pay mechanical royalties to songwriters and publishers on your behalf for both physical and digital releases.
NOTE: This explanation is not intended to be exhaustive, and applies only to audio recordings. Significant additional rules apply to music usage for DVDs and other video instruments, a subject not covered here.