I assume we are talking in an Internet environment.
As the name suggests, client side scripting is carried out (executed) on the client's machine, eg., by the browser. The most common script for this purpose is JavaScript. This language is experienceing something of a renaissance as part of the group of methods for producing dynamic web sites -
Ajax.
It is used for things that are more properly carried out at the client end.
Server side scripting happens at the other end of the line, eg., at the server. This uses languages such as
PHP, Perl, etc.
It is used for such things as producing dynamic content for a page. Providing the interface between the client and server side databases. Processing form information and emailing.