Once you've pressed the embed button, it will reveal all your embedding options. You can choose from several options for your custom embedded YouTube video. The first choice is whether to include related videos. Related videos will ordinarily show up in little boxes on the bottom third of your video player. I find them distracting, so I usually disable related videos by unchecking the box.
The "privacy enhanced" mode has to do with whether or not Google sets a browser cookie for videos when the page has loaded, but nobody has clicked to start playing the video.
You also have the option to include a border to distinguish your video from the surrounding content. If you do not create a border around your video, your color choices will only show up on the bottom player bar. You should see a thumbnail preview to show how your video player will look.
Next you choose from the available color palates, and then choose a size. Choose a size that is large enough to show the video but not too large for the bandwidth of your audience. If you're embedding videos in your blog, often the smallest option is the best. If your video exceeds your blog's column width, it will mess with your blog's page layout.
YouTube automatically updates the embed code as you make choices, so you don't need to close the customize area or press any save buttons. Just copy the embed code from the top of the options box you opened when you pressed the embed button.