Client database applications run on the user’s computer. Often they attach to data which is stored somewhere on the network, sometimes they are self-contained and store their own data. Typical examples would be applications built with Ms Access or Filemaker.
The strong points of such applications are that they are quick and easy to build. The base programs have lots of tools to help develop form, queries and reports. If the data is kept in on application on a network server, different users can run local copies of the application front end on their own machines and all attach to the same data.
The downside of such applications is that they are local. Often people need access to their data from more than one location and while there are ways of synchronizing data between different client databases in different locations it’s much more straightforward to put the database application on the web, where it can be access from any computer that has internet access.