Depending on your needs, it may be easiest to use a DB that requires no installation, like HSQLDB. If you want a "real DB", either Mysql or PostgreSQL would work fine. Googling for "download mysql|hsqldb|postgresql" will get you the download locations.
If you wanted to use A REAL DATABASE, you could download the Oracle Home edition for free. You could also download the development version of Weblogic Workshop 8.1 (free), which comes with a handy-dandy fun-and-easy-to-use JDBC connector (actually it sux, but what do I know?) Then, you could download the CrystalReports 10 that comes for free if you have Weblogic Workshop 8.1. Now, you'll have a production J2EE container, a production database, a production report writer, and possibly... wood? Enough of that nonsense!
Oh yeah, now you can start programming using production tools, and learn why most professional developers have more fun screwing around with the free stuff at home. :-) The free stuff is just kewlerino
Huh, wha? Oh, yeah, or try that MySQL, I've heard keen things aboot it.
If at first you don't succeed, try, try again. Unless you really suck at it. Then, you might just want to try something else, if you dont' want to be a loser I mean.