A quick search on WP came up with some outdated plugins and some with no screenshots.
I specifically need to embed playlists; interested in screenshots if you can share.
The YouTube embed plugin supports playlists, features a screenshot, and appears to be actively maintained. To embed a playlist, install the plugin and then add the following shortcode in the post editor:
The playlist id is the string of numbers and letters after the "&list=" in the YouTube URL for your playlist:
You can also embed playlists without a plugin by pasting the YouTube embed code in your post editor's HTML area, like this.
To elaborate upon Nick's answer, WordPress supports oEmbed since version 2.9—and so, a YouTube video could easily be embedded by pasting its URL in the content area on a line by itself. The URL should be a plain text one (without an
Customization for how the video gets embedded does, however, require use of a different solution.
This information was acquired from the WordPress Codex: http://codex.wordpress.org/Embeds
It should also be noted that YouTube is not the only website which WordPress supports for oEmbed use. A complete whitelist of URLs can be found at the above listed page.
What about "Jetpack by WordPress.com"? It is supported by Automattic (WP creators) and one of its features is "Easily embedded media from popular sites like YouTube, Digg, and Vimeo."
Of course, it has many other features that some may consider as overhead.