DNS is Domain Name Service.
DNS is the internet protocol and service which provides us with domain names, such as yahoo.com and google.com. Without DNS, there would not be domain names. There would only be IP addresses.
Domain names refer to computers; commonly these computers are web servers or email servers. Internet traffic can be sent to a given internet-connected computer by addressing the traffic to the IP address of that computer. When a computer is identified by means of a domain name, the domain name must be translated into an IP address before the computer it belongs to can be accessed. Computers use DNS servers to look up the IP address(es) associated with a given domain name.
Typically, a home user will use the DNS servers provided by his or her ISP. This is generally the default configuration when you set up a modem or modem/router to connect to a particular ISP. However, it is possible to use other DNS servers as well.
There are a number of public DNS servers which can be used as an alternative to your ISP's DNS servers. These include Norton DNS, Google Public DNS and OpenDNS, to name the ones I am aware of (you have just introduced me to Norton DNS; I had previously not heard of it).
Norton DNS appears to offer a security filtering service. Norton DNS is meant to check to see whether domains you try to access have been identified in Norton's database as unsafe. If a domain has been identified as unsafe, the Norton DNS server will, in effect, block access to it.
So, for instance, suppose you type http://www.example.com/some/bad/website.php into your web browser, while using Norton DNS. If you've haven't been there recently (so the domain is not in your computer's DNS cache), your computer will send a request to Norton's DNS servers to ask for the IP address for www.example.com. But, suppose that Norton has identified www.example.com as a rogue website, to be avoided. Rather than telling your computer the IP address for www.example.com, Norton's server will instead tell you a different IP address, owned by Norton. Your web browser will accept this at face value and go on to request /some/bad/website.php from the Norton IP address as if it were the actual IP address for www.example.com. Norton's web server will then send your web browser a web page providing a security warning about the site you tried to access.
This is the basic means of operation for all DNS-based content filters and website blocking software.
Because DNS is such an important service, which affects almost your entire internet usage, it is important to make sure that you can trust the DNS servers that you are using. A rogue DNS server can make any domain name point to any computer it likes. To take an extreme example, the actual web address for your online banking could be directed to a convincing phishing website, such that it would be impossible to access the real thing without changing back to a more trustworthy DNS server.
I'm not saying that you can't trust Norton DNS. I'm just saying that it's worth thinking about the amount of trust you are, by necessity, placing in whoever provides your DNS servers.
One common problem which is found even with legitimate DNS servers (and also with some ISP's DNS servers) is NXDOMAIN hijacking. https://en.wikipedia.org/wiki/DNS_hijacking#Manipulation_by_ISPs
Free DNS servers tend to be funded by advertising presented to you through NXDOMAIN hijacking (which is an understandable compromise). While you might consider this unobtrusive, it is important to note that it can cause unexpected problems with some software, and may create a security concern in some cases.