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###At least for the PHP part:

###For the Javascript part

At least for the PHP part:

For the Javascript part

At least for the PHP part:

For the Javascript part it becomes a question on what and if you need to optimize.

That said jQuery gives you a overhead, but you might/likely do not want to use pure Javascript.

###At least for the PHP part:

###For the Javascript part

it becomes a question on what and if you need to optimize.

That said jQuery gives you a overhead, but you might/likely do not want to use pure Javascript.

Edit:

Had another look at this, and also had to check if I was missing something regards to your variables in the Javascript. But, unless you have some omitted code within the "submit" function – you have some unused variables:

var Email = $('#Email').val();
var Password = $('#Password').val();
var PasswordRepeat = $('#PasswordRepeat').val();

As you use:

data : $(this).serialize(),

Which in effect serializes all form fields with a name attribute, and not something like:

data : $.param({
 Email : Email,
 Password : Email,
 PasswordRepeat : PasswordRepeat
 }),

By which you could also use:

data : $.param({
 Email : $('#Email').val(),
 Password : $('#Password').val(),
 PasswordRepeat : $('#PasswordRepeat').val()
}),

It depends on use. If for some reason your form has lot of fields that you do not want to post the latter would be to prefer, though that would also bind it much tighter together and make it a possible spot for bugs in future changes to the code.

As a side-note you also should most likely have this:

$('#form').on('submit', function(event) {
 event.preventDefault(); // Prevent form submit.
 ...

At least for the PHP part:

You should make it into a function where you return status instead of going over every possibility even if you have a fail.

function validate_post() {
 if ( ... )
 return "no";
 if ( ... )
 return "yes";
}
$array['reponse'] = validate_post();

At least for the PHP part:

You should make it into a function where you return status instead of going over every possibility even if you have a fail.

function validate_post() {
 if ( ... )
 return "no";
 if ( ... )
 return "yes";
}
$array['reponse'] = validate_post();

By this you would eliminate the last // Si tout est ok part as well, as by the response would be OK.

If you truly want to eliminate possible steps, you could also negate the isset() checks and use OR instead of AND, as it then will stop on first false.

if (!isset($_POST['Email']) || !isset($_POST['Password']) ... )
 return "Une erreur s'est produite.";

The length check for password require less resources and as such could be moved above the email check. (Which would be the most expensive.)

For the Javascript part it becomes a question on what and if you need to optimize.

It is likely a process rarely taken. Manual action by user (hopefully) not in a loop ;). When it get submitted you are to have a rather big form for it to have a noticeable effect.

That said jQuery gives you a overhead, but you might/likely do not want to use pure Javascript.