Open-back headphones have the back of the earcups open. This leaks more sound out of the headphone and also lets more ambient sounds into the headphone, but gives a more natural or speaker-like sound and more spacious "soundscape" - the perception of distance from the source.
Closed-back (or sealed) styles have the back of the earcups closed. Depending on the model they may block 8-32db of ambient noise, but have a smaller soundscape, giving the wearer a perception that the sound is coming from within their head. One reason for this is that there are sounds reflected back towards the ear.
I hope that clears it up.
I would say that a Bose AE2i should be good, since it's comfortable and the sound is good, with a decent price. I have a friend who owns one, and she's quite happy with it.
And I'm assuming that your budget is in USD, not HKD, right?