What is Server-side Scripting?
In my own terms it would be defined in the following example: A user wants to search a database located on a server over the internet. The request the user makes, is transferred to a server side script such as PHP which processes the request and returns the result back to the user's browser.
Basically the information is processed on the server side of the house, as opposed to client side scripting where the processing of information is on the user's system.
From my perspective, scripts reside on the server directly, and this is where web developers or designers have the ability to create dynamic elements on web pages through the use of PHP for example.
What is Client-side Scripting?
This is where the scripting is located directly in or on the users website browser page. The client-side of the house in other words, and not the server side. The webpage responds or is altered using Javascript for example based on input from the user.