jQuery fadeIn fadeOut with click

you need the "." before button_contact

$(document).ready(function(){ 
  $(".button_contact").click(function() { 
    $("#contact_form").fadeIn("slow");
  });

  $(".contact_close").click(function() { 
    $("#contact_form").fadeOut("slow"); 
  });
});

jQuery also has a .toggle() function that allows you to pass multi-functions that are toggled between each other when the element/s are clicked.

http://api.jquery.com/toggle/ a nice function because you can add as many functions as you want.

$(document).ready(function(){ 
    $(".button_contact").toggle(function() { 
        $("#contact_form").fadeIn("slow");
    },
    function() { 
        $("#contact_form").fadeOut("slow"); 
    });
});

You forgot a . before the button close:

$(".button_contact")....

Should work.