Distributed computing is the computing paradigm wherein multiple autonomous computers perform a task by communicating via a computer network. In a distributed computing environment, different components and objects of an application can be located on different computers.
Distributed systems have been in existence for long now and their significance has only grown over time. A few examples come to mind. The whole of Google systems such as search, gmail, youtube etc. run in distributed computing environments, wherein thousands of servers handle terabytes of data broken into many small components. In this case, a subset of servers may be running gmail, another subset running google apps and another one running the database servers.
Another example is that of Facebook. Facebook has thousands of servers across data centers spread over various continents. When a user logs in, the request is routed to one of the gateway servers which then routes the request to an app server in the data center. Then the app server could be contacting a different database server to ascertain user credentials, thereby creating a vast network of distributed computing infrastructure.