DNS stands for Domain Name Service and/or Domain Name Servers. The Domain Name Service (DNS) can be thought of much like a directory. If you need to contact someone via telephone but only have his/her name, you can always check the telephone direcotry for their telephone number. The same applies when dealing with the Internet, although this kind of "lookup" often happens in the background, unnoticed.
Most think of domain names such as 1and1.co.uk when it comes to thoughts of the Internet. Dive deeper and you will find each domain name (such as 1and1.co.uk) maps to an address on the Internet, called an IP address. This IP address is a unique address for a specific server where the website files are located. So when you type www.1and1.co.uk into the address bar of your web browser, your computer checks for the DNS records of www.1and1.co.uk in order to find the address of the server where the website files are hosted. Once the address is returned, you computer then makes a connection to the server to retrieve the website. This all happens in the background with most never even realizing the process (which is much better anyway, because IP addresses like 220.127.116.11 are much harder to remember than 1and1.co.uk!).
DNS, just like a telephone directory, contains more information than just a telephone number. A telephone directory will also typically have a physical address listed with the name. So if you want to send two physical letters to John and Janice Smith, and you find they both live at the same address, then the only difference on the two letters would be the addressee. DNS also holds other records, such as the address of the server that handles e-mail (known as the MX record) for a domain. You may write an e-mail to firstname.lastname@example.org and later that day send an e-mail to email@example.com, but since both e-mail addresses are "at" 1and1.co.uk, both are sent to the same e-mail server. John and Janice cannot open each other's mail, but both arrive at the same place.
Some default DNS settings are already configured for you when you register a domain name with a hosting provider. This way your website loads correctly from the host's server and your e-mails are routed to your host's webmail. Almost all providers allow you to alter the DNS settings yourself however, so that you can customize some routing if you choose. This was you can set route your visitors to the IP address of another server that hosts your website files or route your incoming e-mail to another mail server that handles the e-mail for your domain.
This is convenient if you have purchased a domain from a company that has cheap domain registrations but expensive hosting plans. You could purchase the domain from one provider and configure the domain to use the DNS (telephone directory) of the other hosting provider. The hosting provider's DNS will already be configured to point to the right servers for your website files and e-mail, so all you need to do is configure the domain to use a different routing "directory" (DNS) and viola!
Changing DNS servers take time (much like a change in address) and may take up to 48 hours until the update has been recognized by all servers on the Internet. Please plan accordingly when making any DNS changes.