Among it features you will find the possibility to perform secure connections, replication or search and index text fields. It also had a lot of repercussion due to the fact that it was multi-platform and that it had a large SQL language pack.
Since it's open source, many companies adapted it to their own needs and very little time went by before it was accepted by the majority of people in what regards to databases focused on websites. Its InnoDB storage engine and the speed with which it works managed to make MySQL 4 withstand for more than 6 years in a world that is ever changing like the Internet, being launched in 2002, and not being completely substituted until 2009.
If you use content managers like Joomla, Wordpress or PHP-Nuke, you'll probably need a database like MySQL 4 for them to work.