I came down this morning, woke my Mac from its sleep and saw a dialogue box on the desktop which stated:

Another computer is using your IP address. Change your IP address.

Or words to that effect as I can't now quite remember it exactly as it was.

However I have no understanding of how this came about or the significance of the message.

Can anyone enlighten me please?

This sometimes happens. I assume that your computer receives its IP (Internet Protocol) address from a DHCP server. If a network device is inactive, then the server may issue its IP address to another device on the network. The only way to prevent this is to get a fixed IP address from your provider. It will cost you much more. Go to your Network preferences pane and press the Renew DHCP Lease button. This should force your DHCP server to issue you an unused IP address.

Thanks MisterMe.

Just one other thing. My router tells me that between midnight last night and 2.00 am I received a number of Xmas tree attacks (DoS attacks presumably). I don't suppose this could be connected in some way?

I would not assume that an Christmas Tree attack was intended as DoS. The primary use of this technique is to probe your system and its defenses. To your specific question, nobody can know answer it conclusively. However, I would not rule out the notion that you had a duplicate IP address as a result of the Christmas Tree attack.

.... I would not rule out the notion that you had a duplicate IP address as a result of the Christmas Tree attack.
Hmm. Is there any way do you think that I can tell whether my security has been breached?

If I were you I would reset the router to defaults and not use the same password on the router that you used before. Make sure you make the password complicated and not a pure directory word.

Also consider getting a newer router. Some of the cheaper routers seem to go wonky after 3 years.

Little Snitch would be a good way to start...
http://www.obdev.at/products/littlesnitch/index.html