The word Client Side means Active Links as bind to execute with Client/User Interaction.
If certain operation is doing data retrival from server it is again executed after user interaction only.
Thats what mostly active links are used to do validations and so many operations but only after User interaction.
if the methods(objects) are stored in the server side and that is being invoked from the client, passing the trigger and that is being executed in the server side and returning the results to the client, we call it as Server Side Object in Remedy. These are nothing but Filters/Escalations.
if the methods are retrieved from the server and stored in the client side and due to some user actions if those methods are being invoked with in the client side itself by passing parameters and thats gets executed in the client machine(instance- BMC Remedy User Tool or Browser) we call it as Client Side Object. This is basically Active Links in Remedy.
The place where it is executed makes the difference and defines it.
In addition to above inputs, I just want to add response to your query of how ALs are able to retrieve and store data on server:
1. ALs achieve this by using the Set-Fields (Server) and Push fields actions.
2. These actions basically make a call to the AR server to fetch or store data. Usually referred to as back-channel calls.
3. Using too many set-feilds (server) operations on AL side is not recommened as it hits the performance at times (depends on the events where these are used).
4. Another was an AL can get/store server data is by using the Service action. When a service action is invoked, the SERVICE type filters are executed which can get some input params and return some output params back to the AL.
More details can be found in ARS guides.