Computers are grouped together in networks. Networks can be connected to other networks and the Internet is a good example of a network of networks.
A domain or workgroup is like the name of the town you live in or the company you work for. The smaller the organization the more control individuals have and the larger the more likely a central office or fulltime city hall.
In a network the machines are given a common name that is either maintained machine by machine or there is a server (larger computer) that keeps up with the network.
In a smaller network, usually 10 or less computers, the computers all belong to a common workgroup. Each computer is responsible for its own security and related settings. In a small business or home this is an easy way to allow your computers to talk to each other by assigning the same workgroup name to each and the IP addresses are on same physical network (usually your router/modem is a DHCP server and handles the addresses). However the machines talk a lot among themselves so with more than 10 computers it gets slow.
With more than 10 computers you would get a server, normally a larger computer, that will handle the network traffic and security and know who all the computers are on the network. In this case you will use a domain and all the computers use the same domain name. However they do their messaging thru the server which then keeps track of who and where everyone is and less traffic. Domains will work with the largest networks.
To see what Workgroup or Domain your computer is on right click My Computer or Computer and choose Properties. In the box that appears you will see the workgroup or domain name and which it is.