How can I debug why simplest MySQL query returns false?

An obligatory update: as mysql ext is no more, here are answers for two remaining MySQL APIs which I written on my site based on the experience from answering 1000s questions on Stack Overflow:

• How to report errors in mysqli
• How to connect to MySQL using PDO (with the aim of the proper error reporting).

In short, for mysqi the following line have to be added before mysqli_connect() call:

mysqli_report(MYSQLI_REPORT_ERROR | MYSQLI_REPORT_STRICT);

while for PDO the proper error mode have to be set, for example

$pdo->setAttribute( PDO::ATTR_ERRMODE, PDO::ERRMODE_EXCEPTION );

As of the old mysql ext,

To get an error from mysql_query() you have to use mysql_error() function.
So always run all your queries this way, at least until you develop a more advanced query handler:

$query = "SELECT * FROM 'users'";
$result = mysql_query($query) or trigger_error(mysql_error()." ".$query);

the problem with your current query is 'users' part. Single quotes have to be used to delimit strings while for the identifiers you have to use backticks:

SELECT * FROM `users`

In order to see these errors during development, add these lines at the top of your code to be sure you can see every error occurred

ini_set('display_errors',1);
error_reporting(E_ALL);

on the production server, however, the value on the first line should be changed from 1 to 0

Use the mysql_error() function:

$query = mysql_query("SELECT * FROM 'users'") or die(mysql_error());

EDIT: Per Col. Shrapnel's comment: you should never use die() outside of a test environment. In general it's bad practice when writing code that's even intended for production.

Here is some more information: http://www.phpfreaks.com/blog/or-die-must-die