Confused about fighter’s Two Weapon Fighting Style

A fighter in a campaign I am running has the Two Weapon Fighting Style and I am having some trouble with finding details on how it works.

  • Does he have to wield two light weapons? Or can he use (for example) a sword and a battleaxe?

  • Can he strike two different targets within his range for one attack (e.g., sword attacks a goblin, off-hand weapon attacks another goblin)?

  • What exactly is the penalty — is it that the damage modifier only goes to weapon #2 and weapon #1 gets no damage modifier?