Mac OS X 10.5 (Leopard) is indeed no longer officially supported*, and as such retains all security risks that existed since the last security update (2011 I believe), plus all that have been developed since. The actual degree of risk you run depends on a number of variables, like the things you do on your computer, and particularly those that interact with the internet. Another part of the equation is the software you use. Vulnerabilities don't just occur in System software, but in all software, including the apps you use. Most of them will be updated regularly, and these updates frequently address security issues. Software that is no longer being updated runs a higher risk of being exploitable by malfeasants, who'll have had more time to subvert it. Examples include Flash and Java. Most if not all such subversions occur during or after internet access.
Apple's position is to upgrade to newer OS versions, or, if that's not possible as in the case of PPC Macs, upgrade the Mac itself so it can run a more recent OS version and other up-to-date software.
*) This also explains why Leopard threads are found in this particular forum, with other unsupported OS versions. All supported OS versions are dealt with in a separate forum.
