Well, I think it's performance-to-price ratio is unbeatable.
Seriously, though, I've not seen any serious performance problems with most uses of MySQL. But if you're looking at an industrial-strength application where you may have hundreds of tables, some with tens of thousands of rows, and hundreds of concurrent users; well, maybe you should be looking at Oracle or such (and be prepared to open up the purse strings to buy the software and the hardware on which to install it, hire DB consultants to optimize it for your needs, etc.)
If you're talking tens of tables and thousands of rows and tens of concurrent users for a non-life-critical application and without deep pockets for funding, then MySQL can probably work just fine.
Whatever you do, don't use MS Access.