Closed Headphones are constructed in such a way so as to prevent sound leaking out (and by the same token to prevent extraneous sound i.e. blocking, coming in) and are therefore good for recording where you don’t want the artist’s cue bleeding into the mic. They tend however, to be somewhat fatiguing because of this isolation from the surrounding environment.
Open headphones allow extraneous sound in but also leak sound out. They are prefered by many however for mixing as they can be worn for longer periods of time, because they less fatiguing; feel more natural and less isolated, plus they are usually physically lighter.
Semi open, as you might guess, are designed to be somewhere in between these two types.