From WPmadesimple
Adding YouTube videos to WordPress
1. Find the video you wish to embed on YouTube (assuming you don’t already have the video address)
2. Click the Share link (under the video)
3. Under ‘Share this video’, copy the video address to the clipboard.
Notice the tick box option to start the video at a particular position.
4. Log into your WordPress Admin panel and edit the post or page (on your website) you wish to add your video to.
5. Locate your cursor correctly and paste the YouTube video address previously copied in step 3 (as below).
6. Click the Update button to complete. Note that you will only see the address in WordPress but the video will be displayed when the page or post is viewed normally.
7. Test your video by viewing the page or post normally.
This is the easy method!
For more advanced options:
Instead of using the ‘Share this video’ option in YouTube, you can instead use ‘Embed’ which presents you with more control over the video such as the display size and whether suggested videos are shown at the end of play.
Copy the resulting HTML code provided by YouTube and paste it into your page in WordPress. But you must insert the new HTML code using the ‘Text’ editing mode for the page, not the usual ‘Visual’ editing mode.