The first test to see whether you can access the database server is to try to create a database. A running PostgreSQL server can manage many databases. Typically, a separate database is used for each project or for each user.
Possibly, your site administrator has already created a database for your use. He should have told you what the name of your database is. In that case you can omit this step and skip ahead to the next section.
To create a new database, in this example named mydb, you use the following command:
$ createdb mydb
If this produces no response then this step was successful and you can skip over the remainder of this section.
If you see a message similar to:
createdb: command not found
then PostgreSQL was not installed properly. Either it was not installed at all or your shell's search path was not set to include it. Try calling the command with an absolute path instead:
$ /usr/local/pgsql/bin/createdb mydb
The path at your site might be different. Contact your site administrator or check the installation instructions to correct the situation.
Another response could be this:
createdb: could not connect to database postgres: could not connect to server: No such file or directory Is the server running locally and accepting connections on Unix domain socket "/tmp/.s.PGSQL.5432"?
This means that the server was not started, or it was not started where createdb expected it. Again, check the installation instructions or consult the administrator.
Another response could be this:
createdb: could not connect to database postgres: FATAL: role "joe" does not exist
where your own login name is mentioned. This will happen if the administrator has not created a PostgreSQL user account for you. (PostgreSQL user accounts are distinct from operating system user accounts.) If you are the administrator, see Chapter 20 for help creating accounts. You will need to become the operating system user under which PostgreSQL was installed (usually postgres) to create the first user account. It could also be that you were assigned a PostgreSQL user name that is different from your operating system user name; in that case you need to use the -U switch or set the PGUSER environment variable to specify your PostgreSQL user name.
If you have a user account but it does not have the privileges required to create a database, you will see the following:
createdb: database creation failed: ERROR: permission denied to create database
Not every user has authorization to create new databases. If PostgreSQL refuses to create databases for you then the site administrator needs to grant you permission to create databases. Consult your site administrator if this occurs. If you installed PostgreSQL yourself then you should log in for the purposes of this tutorial under the user account that you started the server as. 
You can also create databases with other names. PostgreSQL allows you to create any number of databases at a given site. Database names must have an alphabetic first character and are limited to 63 bytes in length. A convenient choice is to create a database with the same name as your current user name. Many tools assume that database name as the default, so it can save you some typing. To create that database, simply type:
If you do not want to use your database anymore you can remove it. For example, if you are the owner (creator) of the database mydb, you can destroy it using the following command:
$ dropdb mydb
(For this command, the database name does not default to the user account name. You always need to specify it.) This action physically removes all files associated with the database and cannot be undone, so this should only be done with a great deal of forethought.
As an explanation for why this works: PostgreSQL user names are separate from operating system user accounts. When you connect to a database, you can choose what PostgreSQL user name to connect as; if you don't, it will default to the same name as your current operating system account. As it happens, there will always be a PostgreSQL user account that has the same name as the operating system user that started the server, and it also happens that that user always has permission to create databases. Instead of logging in as that user you can also specify the -U option everywhere to select a PostgreSQL user name to connect as.
Please note that if you happen to be the system administrator (hopefully on your personal machine) on a Linux system, many Linux distributions create an user ID called "postgres" or something similar. You need to su into this account from the root account in order to be able to access the system wide installation of the postgresql server for the first time.
I had to do a Google search to figure it out. So I thought I might document it here as well.
I wasn't able to use the steps above to create the database without reading beyond this point. The syntax that I found worked for creating the db is this: CREATE DATABASE "mydb";. Note that I did not use the dollar sign. The double quotes are important to maintain the case, since SQL is case sensitive, or the case will be converted to lower. As usual, the semicolon is necessary to indicate the end of the SQL statement.
To the person using the "create database .." syntax:
You are misinformed and perhaps misinforming too. You are referring to SQL syntax, where indeed the line "create database 'mydb'" will be interpreted as instructions to create a database. In order to use the SQL syntax however, some sort of SQL parsing client needs to be used. Like PHP, or part of a C program parsing SQL queries.
This page however explains creating databases from the operating system shell, where a specific binary called 'createdb' is used to access database creation promptly, without resorting to SQL language.
This is the important difference. Even though 'createdb' did not work for you, rest assured you CANNOT create databases typing 'CREATE DATABASE "mydb"' at Windows 'cmd' or UNIX sh prompt. You are mixing terms here. You yourself have mentioned 'SQL', while it is clear the article does not cover the use of SQL, but instead how to create Postgres databases from generic shell command line (as opposed to some sort of SQL-enabled shell).