Domain Name Servers are systems similar in function to directory enquiries, but completely automated, very fast, and FREE!
The majority of network connections use a series of 4 digits to denote the Internet Protocol address. Your router probably has an IP address of something like 10.0.0.2 or 192.168.1.1 (or variations on those numbers, because those are 'reserved' for local networks, rather than being part of the worldwide set of addresses.
It's a bit like having a switchboard in a big firm. Each firm has an external phone number (unique IP address in the worldwide set) and can have extensions 1001 to 1999 for its internal use. The next firm down the road also has extensions 1001 to 1999 but there's no confusion when extension 1001 in one building makes a call, as their outgoing message to somewhere else in the world will have their unique external number 020-1234-5678 and extension '1001' is invisible.
When you enter a web address in your browser, a message is sent out to do an address lookup and after that is answered the browser will remember the number.
So asking your browser to visit www.mumsnet.com may get answered with 85.92.212.70 while a lookup for www.bbc.co.uk will get 212.58.244.66
I've skimmed over the "how it works" because it has some clever little extras (and I am no expert)...
However, If you imagine that in the telephone analogy you call your switchboard, ask for the BBC and an automatic lookup takes place, then the 'clever bit' is that if you had a fax machine, and wanted to fax the BBC, then the fax machine would 'ask' and be given a number suitable for sending a fax.
With DNS, if you send mail via your ISP, then the ISP mail service will ask for a DNS lookup for the numbers of the mail servers and then attempts to send your e-mail..
It's also possible to use multiple servers to share a load, so if you look at the web addresses on a service like Ebay, sometimes you may see www.ebay.co.uk but at other times you see cgi.ebay.co.uk or shop.ebay.co.uk or contact.ebay.co.uk
So, if you have no way to get requests for DNS checks answered, you cannot access sites without knowing the IP address (and unfortunately, even knowing the IP address does not guarantee that you would see pages you wanted - for small businesses, many different websites may be on the same server with the same IP address.
I hope I didn't lose you way back, but feel sure there could be better explanations online and more detailed technical explanations found on Wikipedia.