Memory error

Occasionally when I try to run scripts, I get an error

[Numberman]

Occasionally, when I try to run PHP programs I get the following error:

Fatal error: Allowed memory size of 8388608 bytes exhausted (tried to allocate 737280 bytes) in /home/httpd/vhosts/**DOMAIN EDITED OUT**/httpdocs/forum/showthread.php(1697) : eval()'d code on line 328

Fatal error: Allowed memory size of 8388608 bytes exhausted (tried to allocate 15 bytes) in Unknown on line 0

Now as far as I can figure out, it means that the memory PHP is allocated is being used up, and more is required (I seem to remember being told that PHP has a maximum amount of memory that can be allocated to its operations, but I can't for the life of me remember where), but there's just as much chance that I'm wrong and this is something else entirely. A one off-error like this wouldn't be an issue, but what worries me is that this is mostly occuring in vBulletin and Invision Power Board with only 3-5 people online at once, so it'll get far worse as my sites get more members.

So, I've got a couple of questions.
1) Am I right in my diagnosis of the problem? If not, then what is it?
2) How can I fix it?

[coopster]

Resource Limits [php.net] should help you get it sorted.