You must make sure that the date format is YYYY-MM-DD on your jQuery output. I can see jQuery returns MM-DD-YYYY, which is not the valid MySQL date format and this is why it returns an error.
To convert it to the right one you could do this:
$dateFormated = split('/', $date);
$date = $dateFormated[2].'-'.$dateFormated[0].'-'.$dateFormated[1];
Then you will get formatted date that will be valid MySQL format, which is YYYY-MM-DD, i.e. 2012-08-25
I would also recommend using mysql_real_escape_string as you insert data into database to prevent SQL injections as a quick solution or better use PDO or MySQLi.
Your insert query using mysql_real_escape_string
should rather look like this:
$sql = mysql_query( "INSERT INTO user_date VALUE( '', '" .mysql_real_escape_string($name). "', '" .mysql_real_escape_string($date). "'" ) or die ( mysql_error() );
mysql_*
functions to write new code. They are no longer maintained and the community has begun deprecation process. See the red box? Instead you should learn about prepared statements and use either PDO or MySQLi. If you can't decide which, this article will help you. If you pick PDO, here is good tutorial. – Madara Uchiha Aug 25 '12 at 8:47