How can I embed a node inside on another node? In Drupal 7 I have a content type created by admin (say content type X), And the other part i have a content type created by users (say content type Y).
Now that is what I do when a user creates content type Y, he can select a node of content-type X to embed node x in node Y.

Without link title node by "References Module" or coding html by "node_embed Module".
You can use entityreference together with display suite and have the display of the entityreference field set to rendered entity. Using rendered entities you can can render any entity with a view mode you need for your display. See this question if you want to create a custom template for the output of that node: How to provide template for particular view mode?