signups table should not be surrounded by single quotes ' but apostrophes `
Therefore
$sql = "INSERT INTO 'signups' ('signup_email_address', 'signup_date', 'signup_time') VALUES ('$_POST[email]','$_POST[date]','$_POST[time]')";
Should be
$sql = "INSERT INTO `signups` ('signup_email_address', 'signup_date', 'signup_time') VALUES ('$_POST[email]','$_POST[date]','$_POST[time]')";