Twin Spell effectively doubles a spell (+4 slot increase). It works just like two separate but identical spells were cast at the same time, so there may be 2 saving throws, 2 caster checks vs SR, etc. Depending on the rolls, you may have both spells succeed, both failed, or one succeed and the other fail. If the spell has an effect other than damage or healing, the effects would not stack.
Chain Spell is more complicated, and I only remember that it turns a single-target spell into a multi-target spell, with limitations on the number of targets, and they also must be close to each other. If the spell is damaging, secondary targets get less damage and an extra reflex to save; if it's not damaging, I think they get a saving throw bonus.