Is there an embedded PostgreSql so that we could unit test our PostgreSql driven application?
Since PostgreSql has some dialects, it's better to use embedded PostgreSql itself than other embedded databases.
Embedded does not necessarily mean it must be embedded in the JVM process. It also does not necessarily need to use in-memory persistence. It should be loaded automatically by the dependency management (Maven, Gradle), so that Unit tests can run on every machine without having to install and configure a local PostgreSQL server.
No, there is no embedded PostgreSQL, in the sense of an in-process-loadable database-as-a-library. PostgreSQL is process oriented; each backend has one thread, and it spawns multiple processes to do work. It doesn' make sense as a library.
The H2 database supports a limited subset of the PostgreSQL SQL dialect and the use of the PgJDBC driver.
What you can do is initdb a new temporary database, start it with pg_ctl on a randomized port so it doesn't conflict with other instances, run your tests, then use pg_ctl to stop it and finally delete the temporary database.
I strongly recommend that you run the temporary postgres on a non-default port so you don't risk colliding with any locally installed PostgreSQL on the machine running the tests.
(There is "embedded PostgreSQL in the sense of ecpg, essentially a PostgreSQL client embedded in C source code as preprocessor based C language extensions. It still requires a running server and it's a bit nasty to use, not really recommended. It mostly exists to make porting from various other databases easier.)