Jennifer's screenshot will only apply if you are using self-hosted WordPress, which you are not, and there is a completely different set of steps for embedding video on a self-hosted WordPress blog.
This is the toolset available to a free WP.com account such as yours (notice the obvious lack of the "embed media" button Jenn highlighted above). The arrow I've added points at the "Insert Video" option, which for WordPress.com users will allow you to embed Google Video, YouTube, or DailyMotion videos simply by pasting the link. Other hosts, such as Vimeo, can be used with WordPress.com, but you'll have to input the shortcode manually according to the instructions for your video host as found here. (Yes, I've confirmed that you can't just paste the URL by using some Vimeo videos.)
You might also find VodPod worth looking into. I haven't used it personally, but it obviously comes reasonably well-regarded if it's mentioned in the documentation.
Using the video upgrade (VideoPress) is another option, but not one I would recommend if you're trying to do this on the cheap. The player is nice and all, but there are so many sites that replicate the functionality...not to mention that, should you ever want to migrate to a self-hosted WordPress blog and take your videos with you, you'll need to keep paying for the VideoPress upgrade on WP.com and use this plugin.