The cloud seems to be used to describe two concepts.
The first is when talking about cloud services like Amazon EC2 or Microsoft Azure or Cloudera. They are software and hardware platforms being offered as services which you can rent. Rather than provision your own infrastructure, you use someone else's over the Internet. Key points of the cloud in this context are dynamic provisioning (the ability to change the number of resources you are using based on capacity requirements), pay-per-use (no need to provision for peak capacity so you don't have to pay for resources you don't need), multi-tenancy (many people are transparently using the same set of resources), and virtualization (so you generally don't need to worry about the underlying hardware of the machines you're using, or who else is using them). It's about infrastructure and software as a service that you can provide or consume.
In consumer-oriented advertising, the word "cloud" is typically meant to represent the idea that your data (and in many cases applications as well) don't reside locally but somewhere on the Internet. This is typically achieved by having client-facing applications running on cloud infrastructure as per the first meaning. This usage is however exceedingly vague and there is a tendency for marketing people to just apply the word cloud to anything that accesses the Internet at the moment, which I'm sure will pass in a few years when the next buzzword comes around.