This is what happens:
After your site was loaded, your js kicks in and binds the click event handler to #relation-contact-button. Then you do whatever and call your ajax handler
$("#datas-user").html("<%= escape_javascript(render :partial => '/pro/users/show_partial', :locals => { :id => @user.id }) %>");
everything inside #datas-user is removed and replaced with your partial and no event listener is bound to anything, since the element which your handler was bound to is gone. You have to either bind your handler to the newly rendered #relation-contact-button again in your ajax success handler, or use:
$("#datas-user").on("click", "#relation-contact-button", function(){
$('#relation-contact').show();
});
instead of
$("#relation-contact-button").click(function(){
$('#relation-contact').show();
See the .on() doc for more information.