You should use IS NOT NULL. (The comparison operators = and <> both give UNKNOWN with NULL on either side of the expression.)

SELECT * 
FROM table 
WHERE YourColumn IS NOT NULL;

Just for completeness I'll mention that in MySQL you can also negate the null safe equality operator but this is not standard SQL.

SELECT *
FROM table 
WHERE NOT (YourColumn <=> NULL);

Edited to reflect comments. It sounds like your table may not be in first normal form in which case changing the structure may make your task easier. A couple of other ways of doing it though...

SELECT val1 AS val
FROM  your_table
WHERE val1 IS NOT NULL
UNION ALL
SELECT val2 
FROM  your_table
WHERE val2 IS NOT NULL
/*And so on for all your columns*/

The disadvantage of the above is that it scans the table multiple times once for each column. That may possibly be avoided by the below but I haven't tested this in MySQL.

SELECT CASE idx
         WHEN 1 THEN val1
         WHEN 2 THEN val2
       END AS val
FROM   your_table
        /*CROSS JOIN*/
       JOIN (SELECT 1 AS idx
                   UNION ALL
                   SELECT 2) t
HAVING val IS NOT NULL  /*Can reference alias in Having in MySQL*/

It actually doesn't accept NULL values it considers it as empty string. That's because you have your server in non-strict mode. That controls how MySQL handles invalid or missing values in inserts and updates. You can read more about modes here: http://dev.mysql.com/doc/refman/5.6/en/sql-mode.html#sql-mode-strict

mysql> insert into cities(state_id) values (20);
Query OK, 1 row affected, 1 warning (0.07 sec)

mysql> show warnings;
+---------+------+-------------------------------------------+
| Level   | Code | Message                                   |
+---------+------+-------------------------------------------+
| Warning | 1364 | Field 'name' doesn't have a default value |
+---------+------+-------------------------------------------+

mysql> select name is null from cities where id = LAST_INSERT_ID();
+--------------+
| name is null |
+--------------+
|            0 |
+--------------+
1 row in set (0.00 sec)

mysql> SET sql_mode = 'STRICT_ALL_TABLES';
Query OK, 0 rows affected (0.00 sec)

mysql> insert into cities(state_id) values (20);
ERROR 1364 (HY000): Field 'name' doesn't have a default value