Mais bon ça met bien que l'erreur est à la ligne 11 du ficher connexion qui renvoie à cette ligne :

  $reponse = $bdd->query('SELECT * FROM joueurs WHERE pseudo="'.$_POST['pseudo'].'"')or die(mysql_error());

<?php
function ConnexionSql()
{
 $mysql_host = "Nom_serveur";
 $mysql_user = "Nom_user";
 $mysql_passwd = "M_passe";
 $mysql_bdd = "Nom_base";

try
{
	$pdo_options[PDO::ATTR_ERRMODE] = PDO::ERRMODE_EXCEPTION;
	$bdd = new PDO('mysql:host='.$mysql_host.';dbname='.$mysql_bdd.'', $mysql_user, $mysql_passwd);
}
catch (Exception $e)
{
        die('Erreur : ' . $e->getMessage());
}

echo 'OK';
}

function connect_pe()
{
 $mysql_host = "Nom_serveur" ;
 $mysql_user = "Nom_user" ;
 $mysql_passwd = "Nom_base" ;

function connect_fo()
{
	include('connection/mysql.php');
	mysql_connect($mysql_host, $mysql_user, $mysql_passwd);
	mysql_select_db($mysql_bdd3);
}

?> 

<?php
session_start();
require('fonction.php');
require('base.php');


if(!empty($_POST['pseudo']) && !empty($_POST['password']))
{
    $bdd = ConnexionSql();
    
   	   $reponse = $bdd->query('SELECT * FROM joueurs WHERE pseudo="'.$_POST['pseudo'].'"')or die(mysql_error());
	   $donnees = $reponse->fetch();
       
    
    $_SESSION['id'] = $donnees['id'];
    
    $_SESSION['Passe'] = $_POST['password'];
    $_SESSION['PasseC] = md5($_POST['password']);
	$_SESSION['Message'] = TRUE;
 
    header('location: index.php');
}
else
{
?>
<h1>Connexion </h1>
<p>Bienvenue sur<?php echo NOM_site; ?>
<form action="" method="POST">
<input type="text" name="pseudo" value="Pseudo" />
<input type="password" name="password" value="Password" />
<input type="submit" name="connexion" value="Connexion"/>
</form>
<?php
}
?>