Ideally you should also be checking to see if $_POST['mail'] is defined because you will get an error / notice depending on your error_reporting level and display_errors.
Updated code:
if (!isset($_POST['mail']) || !filter_var($_POST['mail'], FILTER_VALIDATE_EMAIL)) {
echo "Please enter a valid email address. We want to contact you using your email address. Do not worry, nobody will be able to see it.";
} else {
$mail = $_POST['mail'];
$formcontent = "Email: $mail";
$recipient = "email@example.com";
$subject = "Mail that uploaded picture";
$mailheader = "From: my website";
mail($recipient, $subject, $formcontent, $mailheader) or die("Error!");
echo 'Your email address has been successfully added to your photo !<br>We will contact you later to tell you how to win the $50 000 :)<br><br>';
}