"Cloud computing" is a blanket term meant to do two things: first, to abstract all possible uses of a client-server model behind a single term, as opposed to more specific use cases like "file servers", "database servers", "web servers", "application servers" etc.; and second, to abstract the server architecture itself, in terms of hardware, topology, location, and even ownership.
In a traditional client-server model, which is definitely still in common use today, a client connects to a server that performs a particular job. This server may host a database, or a series of file shares, or a webpage. When the client connects to that server, there is an implicit understanding of the type of communication and data transmission that will ensue between the two computers. There may also be an understanding by the client, or the end user, of the capabilities of the server's hardware, and its limitations. This relatively "tight coupling" between the client machine and its server can pose problems for a sysadmin who needs to take down a server for maintenance; all the applications dependent upon resources provided by this server have to be pointed at another server, and not all applications and infrastructures are designed with this type of redundancy and failover-tolerance in mind.
In a cloud model, the hardware, topology, division of labor, and even number of actual machines involved is all abstracted behind a single endpoint. The analogy could be drawn to a modern "web application", as opposed to older generations of "website" which were more static. We might guess that there is an application server and a DB server behind the scenes, but we really don't have to care; the web server, as part of its job to serve the full application to users beyond the "edge", provides a unified endpoint allowing controlled access to all the data and services provided by other machines behind this front door.
The upshot is that, with a single endpoint exposed to provide the functionality of the application, that's all a client consumer of the application ever has to care about, instead of where to get its data, where to call such and such a remote application process, etc; that means that the administrators and architects of the service provider within this cloud are more or less free to change the machines, topology and other specific implementation details of this "cloud service" without the clients being any the wiser. Facebook could, if it thought it wise, rebuild its entire data storage system from scratch using a different DBMS and all new servers, and as long as the site remained available during the transition, nobody would ever be the wiser; in fact, Facebook did just that, many times, as it expanded from a site hosted from Mark Zuckerberg's personal machine in a dorm room to dedicated hosting off-campus, to server farms in several locations worldwide.