MySQL is a full-blown relational database compliant with the SQL (Structured Query Language) standard. Along with acting as a backend for web sites, it is also useful for productivity applications.
MySQL is frequently used to support web-servers that require back a database. The advantage that MySQL has is that it is free and it is OpenSource. The disadvantage is that if you don't know SQL then compared to FileMaker you are going to find it harder to get in there and get it running.