All you need to do to embed something into a post or page is to post the URL to it into your content area. Make sure that the URL is on its own line and not hyperlinked (clickable when viewing the post).
WordPress will automatically turn that into a YouTube embed when the post is viewed. As of 3.5, Auto-embeds are automatically enabled and there is no setting to turn them off in the WordPress admin pages. Prior to 3.5, there was an “Auto-embeds” checkbox in Administration > Settings > Media SubPanel.
You can also opt to wrap the URL in the [embed] shortcode. It will accomplish the same effect, but does not require the URL to be on its own line and the “Auto-embeds” setting to be enabled. It also allows you to set a maximum (but not fixed) width and height, like so: [embed width=”123″ height=”456″]…[/embed]
If WordPress fails to embed your URL you will get a hyperlink to the URL.