Client Side Operation : It is something which you directly see in your browser window. E.g. When you select an Assignment Group for an incident, the Assigned To field automatically gets mandatory. Here , the operation that you have performed tales place only in your browser window and it doesn't depend on any backend table which you aren't seeing against your screen. Commonly used client side operation are : Client Scripts, UI policy , UI scripts .
Server Side Operation : Let's say you have written a script logic in which all the child incidents are closed once you close the parent incident. Now, when you hit the close button on the parent incident, you can't see the child incident at the same time in your browser window . However, your script logic takes care of the operation which you don't have visibility to just by looking at your screen. Hence, it's a server side operation.Commonly used server side operation are : Business rules,Data Polices, ACLs, Flows etc.
Coming to your question about assignment rules, it's definitely a Server Side operation as the mapping is stored in some data look up table which is being queried every time you change the category/subcategory field.
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