Allow all remote connections, MySQL

As pointed out by Ryan above, the command you need is

GRANT ALL ON *.* to user@'%' IDENTIFIED BY 'password'; 

However, note that the documentation indicates that in order for this to work, another user account from localhost must be created for the same user; otherwise, the anonymous account created automatically by mysql_install_db takes precedence because it has a more specific host column.

In other words; in order for user user to be able to connect from any server; 2 accounts need to be created as follows:

GRANT ALL ON *.* to user@localhost IDENTIFIED BY 'password'; 
GRANT ALL ON *.* to user@'%' IDENTIFIED BY 'password'; 

Read the full documentation here.

And here's the relevant piece for reference:

After connecting to the server as root, you can add new accounts. The following statements use GRANT to set up four new accounts:

mysql> CREATE USER 'monty'@'localhost' IDENTIFIED BY 'some_pass';
mysql> GRANT ALL PRIVILEGES ON *.* TO 'monty'@'localhost'
    ->     WITH GRANT OPTION;
mysql> CREATE USER 'monty'@'%' IDENTIFIED BY 'some_pass';
mysql> GRANT ALL PRIVILEGES ON *.* TO 'monty'@'%'
    ->     WITH GRANT OPTION;
mysql> CREATE USER 'admin'@'localhost';
mysql> GRANT RELOAD,PROCESS ON *.* TO 'admin'@'localhost';
mysql> CREATE USER 'dummy'@'localhost';

The accounts created by these statements have the following properties:

Two of the accounts have a user name of monty and a password of some_pass. Both accounts are superuser accounts with full privileges to do anything. The 'monty'@'localhost' account can be used only when connecting from the local host. The 'monty'@'%' account uses the '%' wildcard for the host part, so it can be used to connect from any host.

It is necessary to have both accounts for monty to be able to connect from anywhere as monty. Without the localhost account, the anonymous-user account for localhost that is created by mysql_install_db would take precedence when monty connects from the local host. As a result, monty would be treated as an anonymous user. The reason for this is that the anonymous-user account has a more specific Host column value than the 'monty'@'%' account and thus comes earlier in the user table sort order. (user table sorting is discussed in Section 6.2.4, “Access Control, Stage 1: Connection Verification”.)

GRANT ALL ON *.* to user@'%' IDENTIFIED BY 'password'; 

Will allow a specific user to log on from anywhere.

It's bad because it removes some security control, i.e. if an account is compromised.

You can disable all security by editing /etc/my.cnf:

[mysqld]
skip-grant-tables