SHOW TABLES and DESCRIBE TABLE are MySQL-specific admin commands, and nothing to do with standard SQL.
You want the:
\d
and
\d+ tablename
commands from psql.
These are implemented client-side. I find this odd myself, and would love to move them server-side as built-in SQL commands one day.
Other clients provide other ways to browse the structure – for example, PgAdmin-III.
If you want a portable way to get table structure in code, you should use the information_schema views, which are SQL-standard. See information_schema. They’re available in MySQL, PostgreSQL, Ms-SQL, and most other DBs. The downside is that they’re fiddlier to use, so they aren’t convenient for quick access when you’re just browsing a DB structure.