DNS (Domain Name System) Server is used to lookout domain names to ip address, it has more purposes, but for the most part thats their function, public DNS providers are free to use DNS servers that provide that function to anyone that needs it.
So since you are online the DNS service was probably provided by you ISP as part of their system to get you online, so why would you like to change your DNS provider from your ISP to a public one?
The main reasons, in degree of importance, in my view are:
1) Security, your ISP probably uses standard off-the-shelf dns software, also DNS although important is not a priority, security tend to suffer first, especially when new tricks and flaws are found, things like dns cache poisoning (injecting dns records that are not from a authoritative dns server), also
2) Performance, every single time you put a new domain name, that name has to be resolved on a dns server for the domain to show up, that means performance is essential, especially comparing to your ISP dns server that is extremely close to you.
3) Control and Customise, the ability to take control of your dns, of using it to make your browsing faster or check it domains are there, if you mistyped, if there is malware on the page you want to visit, or blocking kids from accessing certain kinds of sites.
So the list here follows a different pattern since im going to check the speed of it (from 3 different locations) as well as my personal preference depending on features and overall performance:
1) Google Public DNS (Like: Speed and Standards / Dislike: None)
2) OpenDNS (Like: Speed and Security / Dislike: DNS Hijacking)
5) Level 3 (Like: Speed and Standards / Dislike: None)
4) Comodo Secure DNS (Like: Security / Dislike: None)
1) the checks were made from California (USA), London (UK) and Faro (Portugal) using namebench http://code.google.com/p/namebench/.
2) i could have made a huge list, but i prefer to provide the cleanest most objective one, than a random copy paste list, so i removed lots of other good public dns providers, just cause at the time they were giving me errors and not performing normally, like scrubit dns, cisco or norton safe dns.
3) opendns would have won (just about the same performance as google public dns), but i personally prefer a dns that gives an error when it fails and doesn’t hijack or disrupt sessions by pushing their search when a server is having a glitch or dns issues, but that’s just me.