Forum Moderators: mack
MySQL is a database engine, server, and tools. Think Microsoft Access, SQL Server, Oracle, etc.
MySQL comes with some command-line tools that will let you create databases, populate them, look at them, do complex queries, etc. There are add-on GUI tools available from third parties that add "on top" of MySQL and make it look more like Microsoft Access, etc.
All access to MySQL data is made through a connection to a server. For most MySQL users here, you will probably run my MySQL server on the same computer as your web server. It is entirely distinct from your web server, though, operates on a different port, and uses a different high-level protocol. (e.g. it's own proprietary protocol - not HTTP).
The nice thing about this is that any tool that works with MySQL (such as the GUI tools I mentioned above, or PHP, Perl, etc.) can access any MySQL database on any connected machine.