If you can connect to the database with superuser access, then

SHOW data_directory;

is the shortest way.

If the server is not running and you forgot where the data directory was, then you really have to guess yourself. An operating system specific convention or the shell history might provide hints.

You can also query for it

select setting from pg_settings where name = 'data_directory';

pg_config shows a lot of information including the data directory:

CONFIGURE = '--disable-debug' '--prefix=/usr/local/Cellar/postgresql/9.3.2' 
            '--datadir=/usr/local/Cellar/postgresql/9.3.2/share/postgresql' 
            '--docdir=/usr/local/Cellar/postgresql/9.3.2/share/doc/postgresql' 
            '--enable-thread-safety' '--with-bonjour' '--with-gssapi' '--with-krb5' 
            '--with-ldap' '--with-openssl' '--with-pam' '--with-libxml' '--with-libxslt'
            '--with-ossp-uuid' '--with-python' '--with-perl' '--with-tcl' 'CC=clang' 
            'CFLAGS=-I/usr/local/Cellar/ossp-uuid/1.6.2/include' 
            'LDFLAGS=-L/usr/local/Cellar/ossp-uuid/1.6.2/lib' 'LIBS=-luuid'

This might be a brew-specific thing, though; without brew I don't know what it shows.

On Ubuntu\Debian, try the pg_lsclusters command.

On my machine:

pg_lsclusters 
Version Cluster   Port Status Owner    Data directory             Log file
8.4     main      5432 online postgres /home/adam/db/main/data    /home/adam/db/main/log

If you can't log in (eg, forgot your password) but the service is running, as MacOS X is a posix varient, you can always check ps to see if it's being passed in as an argument :

On a linux box with two postgres instances running:

-bash-3.2$ ps ax | grep postgres | grep -v postgres:
 4698 ?        S      9:59 /opt/PostgreSQL/8.4/bin/postgres -D /opt/PostgreSQL/8.4/data
 6115 ?        S      8:16 /opt/PostgreSQL/8.4/bin/postgres -D /opt/PostgreSQL/8.4/data_sums
30694 pts/9    S+     0:00 grep postgres

Open postgresql.conf, go to the line:

data_directory='/var/lib/postgresql/main/'

is your answer.

Modern versions of the PostgreSQL desktop client for OSX have a convenient "Server Settings" dialog that will allow you not just to list the data directory but open it directly.

Very useful if you experience a random machine crash that leaves behind a lock file.

On OS X 10.8 and 10.9 (not 10.10) with server app installed, in a terminal window, type

sudo serveradmin settings postgres

On OS X 10.9 Mavericks, this is the output I get from that, which includes the data directory.

postgres:log_connections = "on"
postgres:unix_socket_directory = "/private/var/pgsql_socket"
postgres:listen_addresses = "127.0.0.1,::1"
postgres:unix_socket_group = "_postgres"
postgres:log_statement = "ddl"
postgres:log_line_prefix = "%t "
postgres:unix_socket_permissions = "0770"
postgres:log_lock_waits = "on"
postgres:logging_collector = "on"
postgres:log_filename = "PostgreSQL.log"
postgres:dataDir = "/Library/Server/PostgreSQL/Data"
postgres:log_directory = "/Library/Logs/PostgreSQL"

I just tried this in Yosemite (OS X 10.10) and postgres is no longer listed as a service under serveradmin (although postgres is installed).

sudo serveradmin list

does not list postgres

On Windows Server 2012, the command pg_config did not reveal the data directory for some crazy reason. But I found it by looking at the file C:\Program Files\PostgreSQL\9.4\pg_env.bat, which contained this line:

@SET PGDATA=E:\POSTGRESQL

That was the answer I needed.

Generalizing from the 2 top answers:

To show the data directory directly at the command-line:

psql -U postgres -tA -c "SHOW data_directory;"

To also see all other defined paths to files and directories:

psql -U postgres -c "SELECT name, setting FROM pg_settings WHERE setting LIKE '/%';"

Sample output:

          name           |                 setting                  
-------------------------+------------------------------------------
 config_file             | /etc/postgresql/9.6/main/postgresql.conf
 data_directory          | /mnt/pg_ssd_500/9.6/main
 external_pid_file       | /var/run/postgresql/9.6-main.pid
 hba_file                | /etc/postgresql/9.6/main/pg_hba.conf
 ident_file              | /etc/postgresql/9.6/main/pg_ident.conf
 ssl_cert_file           | /etc/ssl/pg_server.cer
 ssl_key_file            | /etc/ssl/private/pg_server.key
 stats_temp_directory    | /var/run/postgresql/9.6-main.pg_stat_tmp
 unix_socket_directories | /var/run/postgresql
(9 rows)

(On Windows, replace LIKE '/%' with LIKE '%/%' because the path starts with a drive letter)