How to make Ajax request from a drupal page?
Have look at jQuery.ajax():

$.ajax({
  method: "POST",
  url: "emailcheck.php",
  data: { email: "test" }
})
.done(function( msg ) {
  alert( "Data Saved: " + msg );
});


Also check your url. Your PHP file has to be in the same folder as your js file when you use relative paths.