I had a domain that I let expire, and was subsequently purchased by someone else. I have several links on my website that are now directed towards their new site. I was wondering if it is possible to find a cached version of the website from when I still owned it, and that way I can save the materials from that page.
Use the Internet Archive's Wayback Machine: http://www.archive.org Note that they only have content from at least six months ago available.
Otherwise, you can try using Google's cache by searching for "cache:youroldwebsite.com" or using the "Cached" link on search results.
The basic model for this is that a domain name points at an IP address, the server machine at that IP address runs web server software, and your web content is on the hard drive of that server machine.
Having a domain expire generally means that "www.mydomain.com" no longer points to the same IP address, but it doesn't automatically cause your content to be erased from the hard drive of the web server machine that it was on.
(Unless, I suppose, you got a package deal with a single company where you could register a domain name and they'd also set up some web server space for you. In that case, they may employ service management software that automatically deletes the contents of old accounts when the domain expires. But you should check with them anyway.)
If you still know the IP address of the web server you had pointed your domain name at, you may be able to get to your content by putting the IP address instead of the domain name into your URLs. You'll probably do even better to edit your "hosts file" (/etc/hosts on Unix including Mac and Linux, and something like %SystemRoot%\system32\drivers\etc\hosts on Windows) to point your old domain name at your old IP address, because fewer links will break that way.
One more thought: On the off chance that your hosting service's DNS server still thinks it's authoritative for your expired domain, you could try using a DNS query tool to specify that you want to ask that particular DNS server what the IP address is for your expired domain. If it gives a different answer than other DNS servers give, than that's probably the IP address of the web server where your content may still reside. You could then configure your client machine to use only that company's DNS servers as your DNS servers, and you'd be able to browse your old site. Quickly use some web site downloading/archiving tool to suck down a copy of your entire website (all the static content at least) before the company deletes it.