All you need to do to embed a video (YouTube or Vimeo) into a post or page is to post the regular 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).
Looks like this when published:
Check out this cool video: That was a cool video.
WordPress will automatically turn that into a YouTube embed when the post is viewed.
For more information on how to embed various media in your posts, please see:
WordPress embeds: http://codex.wordpress.org/Embeds