Do you have work spread over a number of PC's, memory sticks or would simply like to access certain files 24/7 no matter where you are and whether or not you have your own computer with you.
Cloud computing allows you to store data online and access it much like you would your email, so you could take have all of your travel or work docs stored up in the cloud next time you travel or you simply don't have the option of taking your own PC with you.
Even better you can store shared documents online with other teachers or your kids that are instantly updated to all users as soon as they are altered. So no more duplicates and hundreds of versions of the same file on your network.
I use this quite regularly for certain types of files such as my CV, presentations I might want to access but be warned that this technology does have it's limitations. Obviously if you don't have an internet connection you are stranded. And more importantly every time you access these files you are using your net usage - so in schools if you are encouraging your kids to do this you can quickly accumulate a massive internet bill that you really might not have had to, music and video can be painful to get your hands on dependent of your internet speed also.
That said. There are some great cloud computing solutions available to you today for those who are pretty much 24/7 web connected and want to keep everything in one central location.
Windows live Skydrive - offers 25GB of storage and no limitation on file type. Once installed it actually operates much like your traditional file explorer - No bells and whistles and pretty user friendly and pretty basic..
Dropbox - Works fine across both Mac and PC and requires you to install the software locally on your PC and synchronises those two computers. You can invite users to your cloud space this is for the more tech savvy but offers a few more options than skydrive. There is a free option available of this at the moment that works quite effectively.
Windows Live Mesh - Is a cross between remote computing and cloud computing - with similar features to dropbox , but just a little more again. Once you install the software on your computers and login you actually access your cloud desktop and any other of your real desktops as long just as though you are at that terminal. It is still in Beta and has some limitations but this is probably the best look at the future of cloud computing at the moment.
Yes - Apple has it's own product - Mobileme - but it is neither free nor reliable at this point but they are working through a few problems with it all the time. Keep an eye on it.
My best advice is to check these out for yourself. I am sure you will find a regular use for one of them - but don't put your life on it without backing up.