ICQ is a program like Instant Messenger. You have a buddy list and can leave messages for people and all that stuff. Each person is given a number to give out to friends so they can be put on his/ her buddy list.
ICQ (a play on I seek You) is a program that a bunch of young programmers from Isreal put together that allow you to talk instantly with other people that you know that also have the program. You are given a number as well as your friends and this allows you to just talk to them live. All the other instant messanger programs came about because of ICQ. AOL eventually bought them for about $200 million or so a few years ago.
It is a free program and free to use. Since AOL took it over it does have banner ads though.