This is going to be a starting point answer, hopefully you can find something strange while comparing the shots and fix the problem.
If you can connect to Skype that means you have connectivity, but your computer is unable to perform name resolution (that is, convert the address of a service -like a webpage- to an ip address). Skype may work because it relies less on DNS and instead uses its own discovery protocol, or the entries that Skype requires may be saved from previous sessions.
In any case, start by going to System Preferences and typing DNS in the top right corner:
note: the shots come from Lion (10.7.4), so you may see some differences
Now tap on Network (I believe Snow Leopard was the same).
You should be in a page that has multiple tabs, so let's see the important ones:
TCP/IP: let's see if you're using DHCP. If that's the case, then the DNS should be automatically set up for you, but this may not the working (some routers fail to provide this info or provide old info).
There isn't much to see here; I am using DHCP, so my network is automatically configured for me. Let's check the DNS.
Now things get more interesting. I have two that correspond to OpenDNS (which you can use to test if you wish, they are free and open). One thing to note is that they are grayed out, meaning I can't change them. The reason behind this is because DHCP configured them for me. The real reason is because I have configured my router to provide these two DNS addresses to my computers and devices, so we all use a reliable service. But I could override this if I wanted by adding more (notice the plus symbol at the lower left corner). When you press that, you're given the option to type an IP address (use OpenDNS for this test or the Google servers: 188.8.131.52). Add just one, it's just a test.
While we are here, we can check one more thing, Proxies. Go to the proxies tab and make sure that -unless instructed by a crazy network administrator- everything is unchecked, specially the first two. Look at it:
All right, you're all set, go ahead and test your machine by opening Safari and typing www.google.com, performing a search and then testing any of the results.
What to do if this doesn't work?
Despite the above, things can still not work, so don't panic yet. If that's the case take some time to provide us with some details about your network, for example:
IP Address, Subnet Mask, Router, etc. etc. etc.
We can try to see what's going on from there.