One of the great things about YouTube is not only do they allow their videos to be used on other websites, but they even provide the embed code to help you do it!
For an example, we will use the first youtube video ever uploaded: http://www.youtube.com/watch?v=jNQXAC9IVRw. Once you have found the video that you want to embed, first click on Share below the video and then click Embed:
On the Embed screen, it will provide you with the code. It is also recommeded to check the "Use old embed code" box to use the <object> embed code. You can chose the size you want the video below, and also the quality of the video you are embedding if there are multiple options. Once you have chosen the size you want the video to be, highlight and copy the code provided.
Once the code has been inserted on your site, now when someone visits your website they will see the youtube video!