I know a client-side developer write programs for the software inside clients to communicate with the servers(hopefully I’m correct) but what about server-side?Does a server-side developer write programs for the servers to instruct them how to react when receiving orders from clients and so on?
Server side (or backend) developers will usually write the server functionality and handle the database manipulation, they can also make these available for the front-end (or client-side) developer to access through an API.
you can read heaps on wiki
Yes, you just described it perfectly.
Server side: code that runs on a remote machine (or even local if you’re the one doing the development on your pc) and sends/receives data while processing it and potentially storing it.
Client side: the consumer and maker of commands that also sends and receives data; may process it and store it.
The difference is that the server acts as the provider of services and the client as the consumer of said services. Like a bartender and dozens of drunk people.