Every computer (host or router) in a well run part of the Internet has a Name. The name is usually given to a device by its owner. Internet names are actually hierarchical, and look rather like postal addresses. Jon's computer's name is waffle.cs.ucl.ac.uk. We allocated it the name waffle. The department we work in called itself CS. The university it is in called itself UCL. The academic community called themselves ac, and the Americans called us the UK. The name tells me what something is organisationally. The Internet calls this the Domain Name System. Names in this system are "Case Insensitive", which means that it makes no difference whether you give them in capitals or not.
Everything in any part of the Internet that wants to be reached must have an address. The address tells the computers in the Internet (hosts and routers) where something is topologically. Thus the address is also hierarchical. My computer's address is 220.127.116.11. We asked the IANA (Internet Assigned Numbers Authority) for a network number. We were given the number 128.16.x.y. We could fill in the x and y how we liked, to number the computers on our network. We divided our computers into groups on different LAN segments, and numbered the segments 1-256 (x), and then the hosts 1-256 (y) on each segment. When your organisation asks for a number for its net, it will be asked how many computers it has, and assigned a network number big enough to accommodate that number of computers. Nowadays, if you have a large network, you will be given a number of numbers!
Everything in the Internet must be reachable. The route to a host will traverse one or more networks. The easiest way to picture a route is by thinking of how a letter to a friend in a foreign country gets there.
You post the letter in a postbox. It is picked up by a postman (LAN), and taken to a sorting office (router). There, the sorter looks at the address, and sees that the letter is for another country, and sends it to the sorting office for international mail. This then carries out a similar procedure. And so on, until the letter gets to its destination. If the letter was for the same 'network' then it would get immediately locally delivered. Notice the fact that all the routers (sorting offices) don't have to know all the details about everywhere, just about the next hop to go to. Notice the fact that the routers (sorting offices) have to consult tables of where to go next (e.g. international sorting office). Routers chatter to each other all the time figuring out the best (or even just usable) routes to places.
The way to picture this is to imagine a road system with a person standing at every intersection who is working for the Road Observance Brigade. This person (Rob) reads the road names of the roads meeting at the intersection, and writes them down on a card, with the number 0 after each name. Every few minutes, Rob holds up the card to any neighbour standing down the road at the next intersection. If they are doing the same, Rob writes down their list of names, but adds 1 to the numbers read off the other card. After a while, Rob is now telling people about the neighbours roads several roads away! Of course, Rob might get two ways to get somewhere! Then, he crosses out the one with the larger number.