Client side tests are tests of everything that happens on the client side in the broad sense (ie, code that is executed on the clients and might include data that comes from or sent to the server). Let's say a Whatsapp is being tested. Client side tests includes functional tests (click Send sends the message), usability tests (use of the Send is comfortable), UI (everything looks good in different types of cases), data integrity (the information it displays is accurate), security checks (XSS can not be done), end user processes (correspondence with someone), load tests on the client itself, environment testing (mobile - interruption tests), compatibility (works on android new and old versions), management of alerts and more.
For that matter, these checks sometimes cover a server side, but the focus is different. The tests are written according to the cases listed above.
Server side tests ore the ones that occur on the server. Usually there is a lot of updates here: version updates (the server receives a request from the client and decides if a new version is needed, and if so, updates), business logic (eg transferring money from one account to another in a bank application), load (eg, many clients working concurrently), security (encryption, etc.), various mechanisms such as queue management, various calculations performed on the server, session management and more.
The communication between the client and the server should also be examined.
Most of the client's tests are tests through the interface (but not only), The server often uses tools such as APIs, dedicated tools built by the company and even use the client itself (with emphasis on the server).