I use at least two computers, Firefox is installed on both of them (latest version).
Is there an addon that stores my current tabs somewhere on the net, and then restores them on a different machine?
The Mozilla fondation has created "Weave Sync", a firefox plug-in that is able to synchronize bookmarks, tabs, history and password between several computers.
Weave Sync works by syncing your current profile with a server on the web. Unlike Google Sync, you can choose your provider. You can use the Mozilla Fondation servers or your own server (or whatever server you want to).
Xmarks is great for this via bookmarks.
Edit: I just realised you want current tabs synchronised. I don't think Xmarks will be the solution for you.