These headphones: http://www.amazon.com/dp/B005N8W1MO say "Closed design dampens ambient noise", does that mean I won't be able to hear anything around me if I have them on? I'm not sure if I would like that.
Yeah, it is a trade off. You can get either closed or open headphones. Closed ones block ambient sound you might not want to hear, as well as prevents sound leaking from the headphones and being heard by those around you (how well they do this depends on the headphone).
However, open headphones have some characteristics in the audio that it produces that many find preferable. Specifically the soundstage is often better on open headphones, enabling you to get a better sense of depth and location from the headphone.
There are some closed headphones that have characteristics that you would associate with open ones, but they are in a different price category.
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I gotta say I have some of those bose noise cancellation headphones and sometimes its pretty freakly not being able to hear anything around you. Especially if your using that home where is generally probably pretty quiet already. I've been snuck up on a number times.