IP address question

[#1]

Please can someone explain something – as it’s too technical for me – but will be simple for you!
The question is about the computer’s IP.  I know what the abbreviation stands for; but, does the IP number change if you change ISP? 
My computer “crashed” (big time!) on Easter Sunday (no, I wasn’t feeding it Easter Eggs!) and so I’ve had to replace it (Hello Vista!); it also happened to be at the time I was in mid transfer to PlusNet.
Thanks
John 

[#2]

Yes and no.
Unless you have a static IP assigned it will change every time you connect your modem or router.
BUT which IP are you referring to?
If you have a router it's the routers external IP address that's seen by t'internet, your PC will keep the same (internal network) one probably.
If you have a modem then there's only 1 IP address and it will probably change every time you log on.

[#3]

Your IP address will be allocated by Plusnet and yes it does change when you move ISPs.
*HPsauce got in before me.

[#4]

Nothing’s straightforward, is it? 
I had – all new on the same day: computer, a Netgear D834G v4 Wireless Router, and I connected with PlusNet.  So – according to my calculations (hopefully) the IP should now be different to the one on my old computer. 
Thanks for the replies – it’s much appreciated.
John

[#5]

You may still not have the full picture based on your reply above.
There are 2 IP's involved here. Your PlusNet assigned IP (called your Public IP) and your local PCs assigned IP (called your local network IP)
The PlusNet assigned IP is what your router gets given to use when it connects to Plusnet. This can either be a static IP (you always get he same IP when your router connects) or dynamic where you will be assigned a different IP each time you connect, from a pool of IPs available. This IP is normally referred to as your public IP because everyone else on the internet will communicate via it and when you send data out it uses this IP to know where to return the answer to.
Your PC's IP will be different. This is assigned by your router when you switch your PC on via what's called DHCP. The router does the clever part of linking (routing) your PCs local IP to your PlusNet IP and vice-versa using what is called NAT (Network Address Translation) - google this if you want to know more. This IP is referred to as your local network IP because it is only visible to the PCs on internal side of the router (where you plug your PC into) and not from anyone on the internet.

[#6]

Thanks Peter for the info - it’s very much appreciated. 
I think I’m getting there - it’s sinking in!  –  albeit s-l-o-w-l-y! 
John
PS It’s good to be back after a few years absence – and find some names you recognise from those days!