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).
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WordPress will automatically turn that into a YouTube embed when the post is viewed.
You can also opt to wrap the URL in the
</code> <a title="Shortcode API" href="http://codex.wordpress.org/Shortcode_API">shortcode</a>. 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: <code>...
If WordPress fails to embed your URL you will get a hyperlink to the URL.
Can I Use Any URL With This?
Not by default, no. WordPress will only embed URLs matching an internal whitelist. This is for security purposes.
Okay, So What Sites Can I Embed From?
You can use all of these: