This line:

 host    all             all             192.168.0.0/24          md5

will let through connections from IPs matching 192.168.0.X where X is any byte.

The IP address of your error message being 192.168.108.161, it does not match this pattern because 108 is not 0.

To enable addresses like 192.168.X.Y, you'd need a /16 instead of /24 meaning that the first 16 bits only are fixed.

Like this:

 host    all             all             192.168.0.0/16          md5

Don't forget to reload Postgresql. From the official docs, use pg_ctl reload. If that doesn't work, then there are other ways to do it listed in this question.

You need to enter the password of the database user, not the OS user. The "root password" is not applicable. Do you know the password of the database user postgres?

Most systems are set up allowing peer authentication for local connections, meaning a system user can log into the database with a database user of the same name without password. localhost is not a local connection. Its privileges are determined by host entries in your pg_hba.conf file.

Since you seem to be running Postgres and pgAdmin on the same server, I suggest you try a local connection without password. Quoting the pgAdmin manual:

On Unix based systems, the address field may be left blank to use the default PostgreSQL Unix Domain Socket on the local machine, or be set to an alternate path containing a PostgreSQL socket.

Bold emphasis mine.
Of course, you have to run pgAdmin as the system user postgres for this to work.

This related answer on SO has a lot more details and links.