Here's the deal-
"A local server....................without needing to contact a server"
This is the part i don't understand. Would our computer serve as a server or is it something else?
It's a bit awkwardly put, but basically you've got it right.
Note that there is nothing magical about a "server", it's just a computer running certain programs that will serve web pages and listen and respond to requests. You can install the same software on your pc (search for WAMP).
But these tools like Google Gears (when google does something it becomes worldwide news as if they invented the concept, but the truth is other companies have been working on this stuff), work a little differently. As I understand it the basic idea is that when you are offline, instead of sending your requests and changes to the server, they are "cached" in a local database on your pc, and held until you are re-connected to the internet, at which point they are transmitted to the remote server.
So in essence your local pc "acts" like the server for temporary purposes, keeping track of all of your changes and requests until the real web server is available, and may even pre-load data from the server before you go offline, so that it can respond to your requests from within the browser (for example if you are going on a road trip, it might download tons of information ahead of time, and then serve it to your browser as you request it, AS IF you were connected to the internet).