The CD copyright says that you may play the music in private and it sometimes doesn't allow you to play it on a computer. Playing the music in public is subject to royalties, and the American Society of Composers, Authors and Publishers issues licenses and collects the fees. Playing music in public includes playing the CD in a restaurant or bar, playing the music on a website or using the music as background "on hold" music in a telephone system. .
CD copyright does not include the right to make copies in any way. The right to make copies of music is a separate license called the "mechanical" license and it is issued by the Harry Fox Agency for most of the music available in the United States. The license may include the right to make digital copies or, in some cases, the digital copy rights are handled separately. Fees go to the Harry Fox Agency who distributes them to the rights holders.
Buying a CD does not give you the right to perform the music that is on the CD. While you may learn to sing the songs or play the music on instruments, performing rights are negotiated separately with many agencies, publishers or the rights holders themselves. Typically there is a fee per song and per performance. Even if you obtain the rights to perform a piece of music, these rights do not include the right to record your own performance.
The lyrics of the songs on a CD are copyrighted separately and the lyrics writer holds the rights.You may write down the lyrics for your own use but you may not make copies and distribute them. There may already be a licensee who has exclusive rights to publish the lyrics and will insist that all copies must be bought from him. Otherwise you may be able to get a non-exclusive license from the rights holder directly.
The musical score copyright is held separately, usually by the song writer who may have written the lyrics as well. There may also be rights associated with the arrangement of the music. You may write down the musical score for your own use but you may not make copies or distribute the score. Different agencies often hold the rights to the score or you may be able to get such rights from the rights holders themselves.