This is about who assembles the content to be presented.
If it is on the server it obviously has all the work of assembling a text that will be the page to be delivered to the client essentially ready for use.
If it is on the client, a set of pages plus several codes will be received by the client and from there it will only communicate with the server to ask for new information that you need, everything you need to modify and adapt in the presentation of the page takes place in the same client.
There are hybrid cases.
Obviously drawing everything on the client can download a little the need for the server to work. In theory, because depending on the case do several isolated requests can generate so much load that can be worse than render on the server. Not to mention that the developer can do everything wrong and make the situation worse.
Rendering on the client can bring more insecurity, breach of privacy, piracy of code. Of course, most of these things can be solved with some work.
Client rendering does not usually go very well with SEO, even if some people believe that there is no problem, IE, sr usually not very good for websites. It works better and has more advantages in web applications that are quite different.
Amazingly there is no control so good rendering in the client, at least in the current state of technology.
I need to leave, I’ll finish later.
Related: What’s the difference between client-side and server-side code in web development?
– Guilherme Nascimento