CCLI stands for "Christian Copyright Licensing International". It was started in 1988 to help churches with copyright issues. They provide churches with legal coverage for song copies (both in print as well as live sound in video). As they say, "now we offer resources for all your worship and creative arts ministries." They also state that, "CCLI helps churches maintain their integrity and avoid costly lawsuits, while also giving churches the freedom to worship expressively and spontaneously." (Whether it's right or wrong that churches face copyright issues in the first place is another blog), but as today's society would have it, songs used in congregational singing and videos shown in a church setting need to be paid for.
The way it works is each song (that people officially copyright) is given a CCLI number. (They deal strictly with Christian music). Then churches pay a licensing fee to use the CCLI database. This means churches can sing and put slides up of words for any song in the CCLI database. Churches are then required to give credit to each song writer as well as have the CCLI number present on the slides during worship. An extra license is purchased (only $100 a year) if churches want to put a video of their service online with the music.
Sidenote, they aren't kidding with the "International" in their name. They are worldwide. Oh yes, you can find a CCLI number on a Christian song written in Iceland or Namibia if you want.