Cloud Storage offers seemingly unbounded computer storage via the Internet.  Cloud storage vendor typically establish large regional data centers to host the computer storage that they sell on an as-needed basis to others.  Customers can access this storage from anywhere, as long as they have connectivity to the Internet.  Customers purchase as much or as little storage as needed at any one time, and don't need to invest in any computer storage themselves or the infrastructure and expertise to manage it.  Many different vendors offer Cloud Storage today including:  Instagram (stores photos and video for general users), Dropbox (stores and allows sharing of files, such as Excel, Word and PowerPoint), Amazon (S3 and EBS offers storage targeted mostly for other companies either for internal use or to be used for services they offer - e.g. Netflix uses Amazon to store many of their movies).
For more information:

• How Cloud Storage Works
• Cloud Storage at Information Week