The operating system of my MacBook Pro is Mac OS X 10.5.8. What is the compatible browser I can safely download?
Just use Safari. If not you can try Opera 12
Opera 12 Link: Opera 12 Download
Most internet browsers no longer support OS X 10.5.x. I would highly recommend using the built in Safari browser.
However you can find the compatible Google Chrome browser at this address: Google Chrome OS X 10.5
If you do decide to download Chrome here are the instructions for the install.
Delete the current version of Google Chrome
Download the new version
Restart your computer