Most hosting companies forbid you from uploading adult or copyrighted material (especially which you do not own) as this would put them in violation of numerous laws. You will need to read the terms of service for your own cloud hosting before signing up to ensure that your site conforms to their standards.
Typically you are allowed only to host your website or other script such as Wordpress blog on a cloud hosting site and it's not to be used for storage of mass quantities of items such as a video archive which you stream online.
Obviously you would be held criminally liable if you did uploaded software or programs that were infected with viruses or Trojans or in some other way tried to breach the security of the host/server communication or interfere with other clients hosting. Also, monitoring other server traffic on the cloud hosting network is not considered acceptable as it's a form of cyber hacking.
As everyone else said, you could not use the cloud for illegal activities or you could be both criminally and civilly liable. And, it's not so hard to be caught when dealing over the internet, no matter how well you think you are protected. A number of things might be prohibited when it comes to online cloud computing/hosting.
For instance, hosting spoof pages which attempt to trick the user into providing sensitive details that are then used for malicious purposes such as identity theft would be forbidden.
However, for regular site hosting you should do well just by hosting your site (assuming it has acceptable content) and will have no issues with your cloud hosting provider.