I don't mean to discourage plugin authors, but just want to comment that this can also be done via MyBB's Custom MyCodes.
If you want to use MyBB's Custom MyCodes (AdminCP -> Message Filters -> Custom MyCode) for something like this, the following will cater for both YouTube URLs and YouTube IDs. To do so, you can add the follow two custom codes:
Quote:MyCode title: YouTube URL
Regular expression:
\[youtube\]http\://(www\.)?youtube\.com/watch\?v=([a-zA-Z0-9]+?)(&.*?)?\[/youtube\]
Replacement:
<object width="425" height="350"><param name="movie" value="http://www.youtube.com/v/$2"></param><embed src="http://www.youtube.com/v/$2" type="application/x-shockwave-flash" width="425" height="350"></embed></object>
Quote:MyCode title: YouTube ID
Regular expression:
\[youtube\]([a-zA-Z0-9]+?)\[/youtube\]
Replacement:
<object width="425" height="350"><param name="movie" value="http://www.youtube.com/v/$1"></param><embed src="http://www.youtube.com/v/$1" type="application/x-shockwave-flash" width="425" height="350"></embed></object>