From the psql command line interface,

First, choose your database

\c database_name

Then, this shows all tables in the current schema:

\dt

Programmatically (or from the psql interface too, of course):

SELECT * FROM pg_catalog.pg_tables;

The system tables live in the pg_catalog database.

This will list all tables the current user has access to, not only those that are owned by the current user:

select *
from information_schema.tables
where table_schema not in ('pg_catalog', 'information_schema')
and table_schema not like 'pg_toast%'

(I'm not entirely sure the not like 'pg_toast%' is actually needed though.)

I you really need the owner information, you probably need to use pg_class and related tables.

Edit: this is the query that includes the owner information:

select nsp.nspname as object_schema,
       cls.relname as object_name, 
       rol.rolname as owner, 
       case cls.relkind
         when 'r' then 'TABLE'
         when 'm' then 'MATERIALIZED_VIEW'
         when 'i' then 'INDEX'
         when 'S' then 'SEQUENCE'
         when 'v' then 'VIEW'
         when 'c' then 'TYPE'
         else cls.relkind::text
       end as object_type
from pg_class cls
  join pg_roles rol on rol.oid = cls.relowner
  join pg_namespace nsp on nsp.oid = cls.relnamespace
where nsp.nspname not in ('information_schema', 'pg_catalog')
  and nsp.nspname not like 'pg_toast%'
  and rol.rolname = current_user  --- remove this if you want to see all objects
order by nsp.nspname, cls.relname;

You can try to trace in the PostgreSQL log file what pg_dump --table table --schema-only really does. Then you can use the same method to write your own sql function.