I know the differences between client side and server side scripting on a web application. But when it comes to SharePoint what exactly means client side and server side?
The CSOM (and the JSOM by extension) is a Microsoft generated collection of libraries that acts as a proxy to the server-side object model. You can see three different forms:
In SharePoint 2013 it was greatly improved as it got additions for working with search, BCS, taxonomies, workflows and user profiles, and many more. You can read more about this comparison here: http://www.andrewconnell.com/blog/sharepoint-2013-csom-vs.-rest-...-my-preference-and-why#mjGRGibEqwKfks6T.99
How to use CSOM
Advantages and disadvantages of SSOM
This video describes in 16 minutes the advantages and disadvantages of SSOM, comparing 2010/2013 and SharePoint Online. It also shows examples of functionalities and apps where you can deploy CSOM and SSOM.
Have a look also at these great posts:
Difference between Client Side API vs Server Side API
SharePoint Web Part Development - Server Side vs Client Side
CSOM vs JSOM vs SSOM vs REST