If I have a Java app (not web app, 100% local), that I want to deploy. Do user have to install something extra other than the .jar file if I use a database?
Also good point if anyone can say each case of this 3 different databases : 1. Java DB (Derby), 2. H2, 3. MySQL.
None of them, or if you want to install a database to the client, Java DB is an option included with Java SE distro. You also need a client drivers to make a connection to db. Derby also has an embedded database drivers, which you can embed to the application.