Explained quite well here:
http://www.webhostingsecretrevealed.net/blog/web-hosting-guides/switching-web-host/
While this link is from a company which would like you to use their web-hosting services, it's also an excellent guide in general:
http://www.hostgator.com/moving
In particular, note that
(a) many companies (such as the one posting that link) offer services to make things easier for you ; and
(b) unless you opt for 'total downtime' (by switching off the service from your original host before the new one is fully live), there will be a period when some domain name servers are still directing visitors to your old site, while other people are seeing content from the new one. If your shop is well-stocked with everything on offer that might not matter too much (although you'd have to consolidate the two different databases of stock levels after the changeover was fully completed). However if you're selling items where you've only got one or two of each in stock, there's a risk that one site will show an item as still being offered for sale (and accept payment for it) after it's already been sold on the other site.