Method 1:

If you're already logged in to the db server, just copy and paste this.

SHOW SERVER_ENCODING;

Result:

  server_encoding 
-----------------  
UTF8

For Client encoding :

 SHOW CLIENT_ENCODING;

Method 2:

Again if you are already logged in, use this to get the list based result

\l 

A programmatic solution:

SELECT pg_encoding_to_char(encoding) FROM pg_database WHERE datname = 'yourdb';

If you want to get database encodings:

psql  -U postgres -h somehost --list

You'll see something like:

List of databases
           Name         |  Owner   | Encoding
------------------------+----------+----------
db1                     | postgres | UTF8

tl;dr

SELECT character_set_name 
FROM information_schema.character_sets 
;

Standard way: information_schema

From the SQL-standard schema information_schema present in every database/catalog, use the defined view named character_sets. This approach should be portable across all standard database systems.

SELECT * 
FROM information_schema.character_sets 
;

Despite the name being plural, it shows only a single row, reporting on the current database/catalog.

The third column is character_set_name:

Name of the character set, currently implemented as showing the name of the database encoding

Because there's more than one way to skin a cat:

psql -l

Shows all the database names, encoding, and more.

Another way of getting the server encoding (described at https://pgpedia.info/s/server_encoding.html):

 SELECT current_setting('server_encoding');

One can also use a similar select to get other settings, e.g. 'client_encoding'.