YouTube has quietly launched a new feature for the people who want to share their video privately; the new feature lets generate a special URL for your private video and send it to up to 25 people, who, assuming they're logged in to YouTube, just have to click to view. Just go here, click on Edit on any one of your videos. Scroll down to "Broadcasting and Sharing Options," and you should see the privacy settings. Click on Private to unlock the different ways of sharing the video, including the option to generate a limited-access URL.
More info: Help Center