I keep getting a message that my Firefox needs updating, but as I understand it, the system is not supported. What will happen if I don't update? Is it possible to update an old computer like mine?
You would be better upgrading Firefox to Firefox 10.0.2
As for Windows XP I imagine it is still possible to upgrade at present, but you will not be able to do so for much longer.
Does that old PC meet Firefox 10 system requirements as far as the hardware specs?
Pentium 4 or newer processor that supports SSE2
512MB of RAM
200MB of hard drive space
If not, maybe it's time for a new / newer PC?
If so, you really ought to install SP2 and SP3; you'll need to have at least SP2 in the near future for Firefox 12 and later.