Cloud is where the servers live, so the difference between "a server" and "cloud" is scale.
To be a little more specific, "cloud" is a huge server cluster/farm with a generic interface that allows the developers to claim as many servers as they can afford to do any task they want. Once created, the virtual server instance they get can be used for any task, like any physical server, but unlike a physical server you just discard it when you don't have any work to do.
In other words:
Physical server: it's all yours. You pay for it, you maintain it, you use it for anything you want. If you don't have any work, too bad, you still pay the bills. If you have too much work, too bad, wait for the existing servers to handle it or buy another server and install it and when you don't need it anymore, you're stuck with it.
Cloud: someone else has a gazillion servers and they are offering you their processing power, for a price. You create a virtual server, use it to handle your work, usually paying for things like net traffic, disk space, the amount of processing power you need for the time you need them. Not enough work? Discard the virtual server to stop paying. Too much work? Add another instance and pay a little more to handle your workload faster.
What the Xbone will use The Infinite Power of the Cloud for? In theory, anything the developers would want to use a server for, but without having to buy their own servers, only rent them from the cloud. Persistent MMO-like worlds, multiplayer, marketing buzzwords, DRM, stuff like that.