Let's go through this step by step. First, here's your current code, tidied up to be readable:
<?php
$username = "root";
$password = "changedpassword";
$database = "User";
$field1_name = $_POST['name'];
$field2_name = $_POST['password'];
$field3_name = $_POST['email'];
$field4_name = $_POST['sex'];
$field5_name = $_POST['school'];
$field6_name = $_POST['birth'];
mysql_connect(localhost, $username, $password);
@mysql_select_db($database) or die("Unable to select database");
$query = "
    INSERT INTO
            create_user
                (
                    name,
                    password,
                    email,
                    sex,
                    school,
                    birth
                )
            VALUES
                (
                    '',
                    '$field1_name',
                    '$field2_name',
                    '$field3_name',
                    '$field4_name',
                    '$field5_name',
                    '$field6_name'
            )
";
mysql_query($query);
mysql_close();
?>
I've made only two changes (tidied the whitespace, and used _name instead of -name, as PHP variables cannot contain hyphens), but it's already a big improvement. The code is no longer an eyesore. It does not have syntax errors, and it is readable. There are still, though, a large number of problems.
First, you see that we are inserting seven values into six columns. This will be a problem. Fix that by removing the first blank value:
$query = "
    INSERT INTO
            create_user
                (
                    name,
                    password,
                    email,
                    sex,
                    school,
                    birth
                )
            VALUES
                (
                    '$field1_name',
                    '$field2_name',
                    '$field3_name',
                    '$field4_name',
                    '$field5_name',
                    '$field6_name'
            )
";
Now we have something that might actually work. It's painfully insecure, with massive potential for SQL injection attacks, and it won't work on the latest PHP because the mysql_ functions have been removed, but it might actually kind of work somewhere. You wouldn't want to put it into production, but for test purposes, we're getting somewhere.