Russell, from your description this could be a number of things (or even a combination!) - Bios settings, CMOS battery, faulty power switch, dodgy computer power supply, faulty graphics card.
Sounds
very odd that you cannot turn the power off using the switch on PC - are you holding the power switch in for long enough?
What is the specification of your PC - Motherboard, memory, graphics cards, peripheral cards, HD etc.?
Have you undertaken any recent hardware upgrades etc.?
Can you see anything on the screen when you try to boot the system?
I agree with MrJay that the first port of call look at the BIOS (assuming you can see the start up screen). I would also remove the battery for 10 minutes or so. Think it also a good idea to get a replacement battery and clean the contacts.
Assuming that this fails you may want to consider disconnecting and removing all cards and peripherals from your system and see if you can get the system to at least boot into the startup screen to access the BIOS. You can then add components back one at a time starting with HD to see if you can isolate the problem.
Hope that this may be of some help
Cheers
Tony