Think massive storage and computer processing power "out there somewhere" -- massive server farms connect to each other and to the web.
If your internet connection is fast enough and sturdy enough you can stream music, movies, tv... do work, write, and store all your stuff "in the cloud".
What is "sturdy" and "fast" enough? The average smartphone package, in a city, can handle it.
There are lots of pros and cons to consider, but that's the nutshell.
ETA: In my day world -- rural school systems, bandwidth is MUCH cheaper than hardware, and repeatedly updating software on the machines is labor and money expensive. The cloud offers some interesting money saving and resource flexible opportunities.
We're building one in Illinois, managed and shared by K-12 schools across the state. Currently, we're testing it out -- running email, student management and reporting, and disaster recovery for a number of schools.