In essence, Cloud Computing is simply running software off of a remote server over a network connection - rather than running software installed on your own machine and using it directly.
It's mostly hype, something for the marketers to talk about. Corporate environments have long used "Cloud Computing", for running all kinds of applications. For instance, the ability to run Word and Excel through Citrix, or to access internal databases off of a specific DBA server.
The fact that corporates have invested in high speed networks has allowed them to move many applications off of the array of machines that they own, and place them on a specific server. This reduces IT costs, because installs and bug patches are performed on one single machine (or maybe a series of Blades in a large organization), rather than having to spend a few months running around and installing software on every machine.
For home users, the availability of mediocre-speed Internet has attracted coorporate attention, at least for lightweight tasks. One example is Google Docs, where you don't have to install or purchase software in order to do some basic word processing; or to some of the online data backup services that do not require special software or hardware.
Does it affect "us"? Well, in Enterprise environments, Cloud Computing is becoming more and more prevalent, as corporates attempt to reduce the cost of IT, while beefing up on other issues, like security or enforcement of accepted practices (like consistent document formats for everyone to use, etc.).
For home users, the benefits are more sketchy, because our "IT Cost" is low. Like it takes us nothing to download and install OpenOffice - while doing the same in a corporate environement would be quite costly. Even if the process is 15 minutes - a tech can do only 4 machines an hour, and if a corporation has 4000 machines, that it 1000 hours. It would take a single tech a half a year to install OO on every machine. On a Blade, it is like a half hour, register the licenses, and everyone is going all at once. But for home users, it is easy-peasy.
Think of other things. Like every home user will have a CD-ROM of some sort (CD, DVD, BR or whatever) because they are cheap - like $25 and you are ready to go. Then there is the Corporate, where the same $25 is multiplied by the 4000 machines, and you have to have someone install them, or purchase them already installed as cost plus - and then some donkeys will inevitably break them by whatever means - so that $8000 Blade server starts looking to be real cheap, fast.
But to make it easy, Cloud Computing is basically running applications from remote servers - and not having to install it or maintain it on your own machine. There are benefits, but also downfalls, so it depends on your needs and requirements, and the availability of a decent network connection that is not capped and throttled, because having a very low cap will make Cloud Applications more costly to run in the long run, over just buying a commercial package in the first place.
As for backups - it comes down to how much you value your data, whether you need "security" or not, and if you feel comfortable saving your stuff on the "Cloud" or not. Again, this is something where Enterprise has a leg up, because they can put in special secure servers that are encrypted and in a physically secure room, and can justify it as a cost savings over having to purchase a myriad of backup media and IT people to back up individual machines - while the Home user can easily purchase a stack of DVD-Rs or keychains or external hard drives at little cost.
It is my opinion that Cloud Computing will be the norm in Enterprise, if it has not already; while for the home users, it will be more specious, and perhaps limited to a few very lightweight packages that are there more for convenience (and the ability to do things you want on machines you don't own, or to share stuff with a Netbook or whatever), rather than becoming the norm simply because we do not have fast, low latency networks that are needed for Cloud Applications to be worthwhile.