ill try to make it short.....
sealed box - tight, accurate sound. 10s and 12s in (THE RIGHT!!!) sealed boxes will cope wll with most types of music, particularly rock, for example, as the instrumenmts used will create a punchy, quick bass sound.
ported - have a hole, a port, in the box. they will ofer deeper bass as can b tuned to offer exrtension at a particular frequency, say 45hz. suited to rap music, for example, as the bass lines are longer and 'heavier' than rock music, for example
free air - no box needed
bandbass/isobaric boxes - lets not worry with these, just for now, will come back to it.
boxes can be made from a wide variety of materials, most commonly MDF, but can also be made from perspex, glass, metal, u name it. in sealed boxes especially its v.important that there is a complete seal so its air tight.
different subs require diffrenet box sizes and its important to get the box right - a great box can even make the dirtiest sub like one of the best.
if u are stratinmg out and want to try your hand at box building, i woukld stick to mdf for now.
there's plenty more to add - but I gotta go 4lunch!!!! Im sure some of the others will halp u out.