Easy MySQL question from newbie

I'm creating a ColdFusion Web applicaiton and need to create a database to use with it. Someone recommended MySQL. My quesiton is, is MySQL a database server or a database application, or both?

I'm on Mac OS 10.4 and want to be able to create a database and test the database locally. I already have the ColdFusion test server running on my machine. Can I just use MySQL to create a database, or do I need a MySQL server running on my machine too?

Any idea where I can download the right software on this huge site?

Thanks,

Bob