You cannot use two different functions on the PSP simultaneously. That means that no music and web browsing. I believe music and pictures might work, but that's it. While the browser is open, all other functions stop. If you could manage streaming audio and another site, it might work. However, the browser has some terrible limitations.

Don't be silly! Of course you can listen to your PSP Radio while using your PSP's built in browser.
Step 1.) Turn on the PSP
Step 2.) Open the Browser
Step 3) Make Google.com your home-page
Step 4) Open your favorite Radio player
Step 5) Hold "Square" and click "R"
Step 6) Select the home-page (aka Google)
Step 7) Have at it!

Upgrade to non OFW.
Install music.prx plug-ins.
Now you can listen while browsing or playing games.

Yeh, you can hear PSP Radio while using the PSP built in browser like Yutsud said.
But if you want to hear the musics that are on your memory stick, you are going to have to use a non OFW, like eddy1nsane said and install the music.prx plug-ins.

What is non OFW and do you have to have a modded psp to have it? cause i would like to listen to music while playing games.

ofw=original Frimwire, and you need a pandora's battery and magic memory stick to have mod/hack

Nandale

Nope you can't. Unless you hack your psp.