How does trello.com keep all the users screens updated with new information as it comes in?
For example, if I change something on my screen, it automatically changes on all the other users screen who are also on that page…Are they just constantly refreshing the page or is their something else going on?
Original Answer
Whilst I do not work for Trello, from the outside it looks like it is using AJAX polling to look for updates to the presented information from looking at the site in firebug.
You can also use Comet to achieve this.
Updated
So I have done a little more digging and I found this on Hacker News:
And then later:
I can only assume that ianthehenry is working for FogCreek, but I am unable to confirm it.
He also goes on to mention some of the other projects they use with:
Update Again
Trello themselves have just written a blog post confirming their stack: http://blog.fogcreek.com/the-trello-tech-stack/