The computer is a 2-year-old Packard Bell laptop running Windows 7 Home Premium on a 2.27 GHz processor with 3GB of RAM.
It just took over 20 minutes to start up, and when it eventually does run, it erratically slows down at little apparent provocation.
I have checked it for Viruses and Malware (full scans with up to date SUPER Anti-Spyware, MalwareBytes and AVG). There are no programs running without permission, or on Start Up. There are no unexpected processes using huge amounts of memory or CPU. The HDD is about 2/3 empty, and I defragged it anyway.
Based on an answer to a similar question here, I tried running PerfMon, and I am seeing spikes of HDD usage when the system slows down, but it's too slow to really see what's going on until after things are running normally again!
What is wrong with this laptop?
Edited to add:
It does seem to have been the Hard Disk - the laptop now won't boot, claiming there is no operating system. Going to take it into the friendly local shop to sort it out. Thanks to all for their advice and assistance.