I won't release my software source code, and it will be a commercial application. Can I use the MySQL within paying any fee to MySQL? if it is illegal, any alternative database suggest?
MySQL itself is open source and can be used as a standalone product in an commercial environment. If you're running mySQL on a web server, you are free to do so for any purpose, commercial or not. If you run a website that uses mySQL, you won't need to release any of your code. You'll be fine.
mySQL has enterprise licenses with (I think) a different code base, and premium support by Sun, but those are entirely optional.
There are limitations on redistribution of mySQL within a closed source product, and linking against mySQL libraries as was pointed out in a different comment. As for redistribution:
OEMs, ISVs, VARs and other distributors that combine and distribute commercially licensed software with MySQL software and do not wish to distribute the source code for the commercially licensed software under version 2 of the GNU General Public License (the "GPL") must enter into a commercial license agreement with Sun.
if you are looking to redistribute mySQL along with a commercial product, check their legal page. I think most companies circumvent this by installing the mySQL server separately.
GPL and linking against client libraries?
I don't know what the fact that the GPL (the license mySQL is distributed under) forbids linking against closed source software means for applications that do not link against mySQL directly, but ship with mySQL client libraries. Do those have to be Open Source? If anybody would like to shed a light on this, in a separate answer or a comment, I'd be most interested.
Yes, and you don't have to release your project's source (but you do have to release any modification you make to MySQL itself (only those not the whole program) if you also release your project in binary form containing the modified MySQL code)
In more simple terms, if you modify MySQL and those modifications end up being distributed in binary form you have to publicly release the aforementioned modifications.