The cloud keeps one copy of the document. When you make a change on your browser, you're actually asking the cloud to make that change for you. Imagine you had 6 people in a room shouting out the changes they wanted, and one scribe was making them all - that's basically how it works. It can be a bit chaotic, but the scribe (the google cloud) is very good at reacting quickly to what people shout out, so it works pretty well.
When you're editing as well as seeing where your caret (I-bar, insertion point thing) is, you see the caret positions of the other editors too - each one is a different colour and shows the name of the editor. Their changes show up on your screen as they make them. There is a bit of lag, but this doesn't normally get in the way too much. There is also an IM chat panel where the document editors can discuss changes with each other.
It''s the sort of thing that fries your brain a bit if you're used to the local files approach, where you make changes over several hours and then have a sync problem. Once you assume everyone is connected all the time, new things become possible.