The multibit.org server is (primarily) used to host the website and enable people to download the MultiBit installers.
Once you have installed MultiBit on your local machine:
+ your wallet(s) is/are stored locally (multibit.org has no idea what wallets you have, nor what passwords you are using)
+ MultiBit connects to the regular bitcoind nodes in the Bitcoin network to receive blocks and transactions. Also to send transactions.
When MultiBit starts up it connects to multibit.org to get a 'is there is new version?' file. At the same time it also sends multibit.org it's current version number. This is how you get the little messages when a new version is available.
If multibit.org was shutdown:
+ no-one would be able to download new MultiBit installers
+ the help files would not be available (as these are also on multibit.org)
+ the 'new version' messages would stop.
But your wallet would sync as normal and you could receive and send bitcoin no problem.
The multibit.org server is a dedicated server hosted in Strasburg, France and (so far) has proved reliable.
When we roll out MultiBit HD we will probably also donate the
http://bitcoin-solutions.co.uk bandwidth for downloads too. (This is another 100 Mbit/s in addition to the current 100 Mbit/s).
If the multibit.org server is hacked then the biggest problem would be if the genuine installers were replaced with a Trojan wallet stealer. This is why the installers are signed with my PGP key and the release notes are always bitcoin signed with the MultiBit donations private key. These keys are not stored on the multibit.org machine.