Replication
Sounds like you need database replication.
There are several ways to do this with Postgres, one built-in, and other approaches using add-on libraries.
Built-in replication feature
The built-in replication feature is likely to suit your needs. See the manual. In this approach, you have an instance of Postgres running on your primary server, doing reads and writes of your data. On a second server, an entirely separate computer, you run another instance of Postgres known as the replica. You first set up the replica by doing a full backup of your database on the first server, and restore to the second server.
Next you configure the replication feature. The replica needs to know it is playing the role of a replica rather than a regular database server. And the primary server needs to know the replica exists, so that every database change, every insert, modification, and deletion, can be communicated.
WAL
This communication happens via WAL files.
The Write-Ahead Log (WAL) feature in Postgres is where the database writes all changes first to the WAL, and only after that is complete, then writes to the actual database. In case of crash, power outage, or other failure, the database upon restarting can detect a transaction left incomplete. If incomplete, the transaction is rolled back, and the database server can try again by seeing the "To-Do" list of work listed in the WAL.
Every so often the current WAL is closed, with a new WAL file created to take over the work. With replication enabled, the closed WAL file is copied to the replica. The replica then incorporates that WAL file, to follow the same "To-Do" list of changes as written in that WAL file. So all changes are made to the replica database exactly as they were made to the primary database. Your replica is an exact match to the primary, except for a slight lag in time. The replica is always just one WAL file behind the progress of the primary.
In times of trouble, the replica serves as a warm stand-by. You can shutdown the primary, then tell the replica that it is now the primary. You can even configure the replica to be a hot stand-by, meaning it will automatically take-over when the primary seems to have failed. There are pros and cons to hot stand-by.
Offload read-only queries
As a bonus feature, the replica can be used for read-only queries. If your database is heavily used, you can offload some of the work burden from your primary to the replica. Any queries that do not require the absolute latest information can be shifted by connecting to the replica rather than the original. For example, a quarterly sales report likely does not need the latest data stored in the active WAL file that has not yet arrived on the replica.
Physical replication means all databases are copied
Caveat: This built-in replication feature is physical replication. This means all the changes to the entire Postgres installation (formally known as a cluster, not to be confused with a hardware cluster) is copied to the replica. If you use one Postgres server to server multiple databases, all those databases must be replicated – you cannot pick and choose which get copied over. There may be alternative replication features in the future related to logical replication.
More to learn
I am being brief here. The topics of replication, high-availability, and disaster-recovery are broad and complex, too much for an Answer on Stack Overflow.
Tip: This kind of Question might have been better asked on the sister site, DBA.StackExchange.com.