We call the computers with a copy of the list of all transactions (the written chain of blocks) a node. Why do we call them such? A node is the name of the points of a net that are connected to each other. Picture a spider web. It’s a bunch of lines connected to each other. The points where those lines touch each others are the nodes of that net. A decentralized computer network is just like that web. Every computer is a node, and connects to one or several other computers. If you see a map of that network, and the network is decentralized enough, it would look like a spiderweb knitted by a spider on drugs.
So those computers will basically do two things:
They will look around for other computers with list copies. If they see that the rest have longer lists (lists that have more blocks written on them), they’ll assume they’re behind and update their copy. We can assume that if a list is longer than another, and their blocks are properly chained, then the list is newer than the others. Think of it for a minute. We just found a way to know which list is older and which new without using clocks or dates written anywhere. We invented our own clock.
The second very important task they do is allow the users to connect to them to check the list. What good is a distributed public list if no one can take a look? So our Monero app will connect to a node to get access to the list of transactions, know if the users received any new coins, and broadcast any transfers of coins that they want to do. After all, a transaction that is never shared, is like the tree that falls down in the forest when nobody is around to listen it.