Typically, your DNS servers, the addresses your computer contacts to find out the IP address of any web site you want to contact, are given to you by your Internet Service Provider. They are the DNS servers that they host, and that the vast majority of their customers use. But there's many other options out there, and it's possible for you to change those addresses. It can sometimes be beneficial to do so.
Why keep your current DNS
Before going into your computer configuration, let's see why you may want to keep your current DNS servers. First, since they are hosted by your local provider, they will typically be faster. So for speed, you may want to keep them. Also, most DNS servers cache results. That means if your neighbor tries to connect to www.cnn.com and then you try to reach that same site, then your ISP will already know the IP of that site and won't need to go out and get it.
Popular DNS servers
If you decide you do want to change DNS, there are several popular servers out there. Google offers a public DNS service. Its servers are at 220.127.116.11 and 18.104.22.168. They claim that their servers use different technologies that make them faster than your ISP might be.
Also, they keep up to date with security updates, which your ISP may not. That means if someone tries to spoof a site name, and you rely on your ISP to tell you what addresses sites are, that may be dangerous. Just imagine if someone could make it so any time you go to www.paypal.com it would transparently resolve to www.badsite.cc instead.
Another popular server is 22.214.171.124 which is now owned by Level3, but used to be owned by one of the very first Internet backbone, BNN. Both this and Google are good choices if you wish to try another DNS, either for testing, to see if they are faster than your current ones, or perhaps to bypass filtering done by your ISP.
Another popular alternative DNS server is a service called Open DNS. Unlike the previous servers which are simply alternative servers available to you, Open DNS offers a range of other services. They offer parental control, optional filtering, and increased anti-malware protection.
They offer two levels of service, OpenDNS Basic and OpenDNS VIP.
Changing your computer settings
If you decide you do want to update your DNS server to another one, then you need to change the configuration of your computer. If you're on a Windows PC, that is done in the Control Panel, in your Network and Sharing settings. Google has a detailed explanation for all the popular operating systems.