DNS (or Domain Name Servers) is basically a system to convert domain names into IP addresses. Domain names are easier to understand, memorize and write for humans while computers only use IP addresses to communicate.
Normally, the ISPs provide their users with their own default DNS server IPs which are often unreliable and slow. There are many better, faster, safer and more reliable public DNS servers for your everyday use. Here is a list of all such fast & free public DNS servers along with their IPs.
Free Public DNS Servers List
1. Google DNS
4. Level 3 Communications
7. Public Name server IP address
If you are wondering which DNS server is the fastest and works best for you, check out this post
Which is the Fastest DNS?
If you are on Windows 7, check out this simple guide to change Windows 7 DNS settings
Easiest Way to Change Windows 7 DNS Settings
And finally, a simple tutorial on how to clear DNS cache
How to clear DNS Cache in Windows, Linux & Mac