Domain name is like your name
A domain name is a 'name' to identify a website, similar to your own name that identify yourself to your friends or a shop name that identify itself as an entity in the street etc. For example, your name is 'Alex' and your friends will call you 'Alex' when you meet up with them, and when your friends gossip among themselves and words like 'Alex has....' or 'Alex is....', they will know it is referring to you, or anyone mentioning 'Jim's bar' will know that they are talking about the popular hang out place down the street etc..
In the internet world, domain name is a unique name for people to identify a website, like yahoo.com is a popular domain name to an internet website withe lot's of fun thing, or google.com is a website to do internet searching etc. While anyone can register a domain name, there is an authority (iana.org) governing the creation of these domain names and there are some rules they have set for people to follow when dealing with domain names.
What are there so many domain name extensions ?
Like giving a name to a new born baby, you would generally choose names like 'Mary', 'Lucy' or 'Stephanie' etc. for a baby girl, and give a baby boy a name like 'Alex', 'Ken', 'Ben etc. or include a 'restaurant' to a shop name that serve good food, a 'bar' for serving beer and cocktail etc.. Similar to this, there are some general guidelines for domain names too. For example, you may have noticed there are lots of commercial domain names ended with a '.com' (yahoo.com, google.com, msn.com), non-profit body ended with .org (redcross.org, wwf.org), government agencies usually ended with .gov etc. This top-level domain names (i.e. .com, .org, .net, .gov etc) was initially used to distinguish the type of websites, it still serve the purpose but has loosen a lot as the internet has been growing so fast and domains names is drying up. To address the domain name shortage, there are more top-level domain names being introduce, we are seeing new top level domain names like .biz, .info, .name etc. being introduced and more are popping up, there is a list of currently available top-level domain names if you are interested to go through the list.
Identifying a country specific domain name
Apart of the top-level domain names, there are domain names ending with .co.uk, .com.mx etc. what are they ? these are country level domain names and are used to identify websites within the country, for example, .co.uk refers to a commercial website in UK, .org.uk refer to a non-profit website in UK, .gov.uk refers to a government agencies in UK etc. Some countries have strict rules on registering domain names in their country and generally requires a local registered business license in the registration process but some countries are loose in this. There is a complete list of ccTLDs if you are interested to check through.