there are two copyrights involved with the duplication and performance of music:-
1. the Mechanical Copyright is to do with the physical copying of the recording of the music onto your medium, be it CD, tape or whatever, and
2. the Performance Copyright is to do with the performance and/or broadcast of the music in public places.
In both cases a royalty is payable calculated on the amount of music copied/played based on time, usually in 30 second increments.
Notice that the copyright is different for both cases, the Mechanical copyright is on the recording and is held by the people who own the recording, usually publishers and/or record companies, and the Performance copyright is on the actual music, usually held by the Composers, (it is infact illegal to sell the composers/lyricists copyright by common law although it still happens in developing countries).
Publishers are individuals/companies who act on behalkf of the composers/lyricists to maximise exploitation of the music as a product in return for a share of the profits.
I imagine you would be involved in the mechanical copyright, and what you will have to do is contact both the publisher of the music and obtain permission to use it.
Even then there are two types of music publishers:-
Commercial Publishers who promote retail music such as pop songs/orchestral works etc,
Production Music Publishers, who promote non-retail music specifically written for Film/Advertising and the likes, also called "Library Music", "Mood Music" or "Drop Needle".
To find out who the publishers are, my first step would be to contact the local Royalty Collection agency in your country. Most countries have only one agency but some have two, one for each type. For example, in Australia it is AMCOS for mechanicals and APRA for Performance. Then there is the US where there are three agencies just to confuse matters worse..
Check out this page [apra.com.au] from APRA which lists agencies/societies around the world. Some have links, others will need to be looked up in the Phone book.
Then ask them about the particular music you are interested in and the will point you in the right direction.
There is a good primer on copyright here [apra.com.au], with addional information at these URLs,
As to performance roylaties of music played over the internet, check here [apra.com.au].
Hope this helps,
added - Ha!, Marcia got there before me!