From what I got from my own frenzied research, here are the pro's and con's of open and closed headphones (I'm not sure what semi-open would be).
open: If you want the best sound quality, get these. However, they do not provide any sound isolation (you can hear everything around you), so if you're somewhere noisy, you'll probably end up turning the volume up to drown the other noise out. Another thing to consider is that everyone around you will be able to hear what you are listening to, so open headphones may not be the best in an office environment, for example.
closed: They are not supposed to sound quite as nice as open headphones (but still pretty good, depending on the pair). However, they keep external sounds out (to various degrees), and others will not hear what you are playing.
When I was buying, I had a budget of around $100 cdn. The two pairs that I ended up considering were the sennheiser 280 pro's, and the akg's 81DJ. Originally, I was set on the 280's, but I found them quite ungainly, and maybe a little too large to carry around all the time. Also, I've heard comments regarding their lack of bass, (see the second link that I posted above for lot's of reviews, comments...), and they would of cost me around $150 after delivery costs. Overall though, comments are quite positive about the 280's. Eventually, I came accross the 81DJ's, which are pretty new (so they don't have as many reviews out). Eveyone seems to love these "cans" and the price was perfect ($100.01 cdn shipped). They are also smaller ("supra aural", instead of "circum aural"), and some people think that they are just as good as way more expensive headphones. So I went with them, and I really like them so far.
That's all that I have to say! Good luck!