What type of crash are you referring to? An app crash (like Word or iPhoto) or an entire system crash?
e.g. a kernel panic as described at…
http://support.apple.com/kb/TS3742
A black screen that comes back up could just be an app crash?
It could be hardware that is failing so make sure you have a backup of the machine on an external disk, just in case it stops working. A good backup will let you troubleshoot safely.
It could also be years of accumulated software running on hardware that is overtaxed.
Some things to consider …
Have you got the original Apple grey install disks? These will have a disk with 'Apple Hardware Test' & instructions for booting into the hardware test written on the disk. Run the test & it may indicate something is wrong (or it may miss any errors entirely).
Boot into the most recent 10.6 install disk you have & repair the boot disk & verify the permissions. You have to click through a few screens to get the top menu to appear so you can select 'Disk Utility', just avoid doing the actual install 🙂
You can try to diagnose the possible fault via a process of elimination…
1. Try a safe boot, does it still crash?
Yes indicates a possible system fault
No points to something in addition to the system.
http://support.apple.com/kb/HT1564
2. Create a new user account & see if that user also crashes when you try similar tasks (it helps if you find a way to cause the crash, but don't keep doing it, crashes can damage data).
3. Generate an EtreCheck report to see if there is any software that looks out of date & can be removed.
http://www.etresoft.com/etrecheck
Post it here if you want some feedback.
4. Backup & reinstall the OS. Hopefully you can avoid adding everything back so that you have a little more free disk space, as disks are more full they get slower to read & write.