There are a couple of ways to do that. The choice will likely be dependent on what you have, and what else you might need to do with it.
First, with your original Clip Imported into your Project, and on your Timeline (do this in Timeline View Mode), just adjust the WAB (Work Area Bar) over the section, that you wish to Share/Export. This article goes into more detail: http://forums.adobe.com/message/3762492#3762492
Or, you can just use the Scissors/Razor Tool, to make any necessary Cuts, then Delete the portions that you wish removed, closing up any Gaps. Then, just Share/Export that edited Timeline.
For YouTube, this FAQ Entry might be useful: http://forums.adobe.com/thread/623549?tstart=0