"Casket" is just an Elizabethan word for a box. Each is an ornate metal container with a hinged lid.
The logic is that Bassanio, who has nothing, chooses the box that is worth nothing, and that's the right box. He wants Portia because he loves her. Morocco chooses gold because he loves riches and believes that gold represents Portia and all her wealth. Aragon chooses silver that is money, which again represents Portia. Bassanio refuses gold because he knows that "all that glisters is not gold." He also refuses silver, calling it "common drudge 'tween man and man," as coins are. He chooses lead because he knows that true worth lies inside, even if the outside doesn't look like much. The lead is more like HIM than like Portia.