You reciprocate when you return a favor, return a compliment, or respond "the same to you" to the angry guy in the car you just passed. In short, you react to an action, statement, or emotion by mirroring it.
This one comes from the Latin verb reciprocare, meaning to move back and forth. Its root makes it sound as if reciprocate implies only a physical action, but it's also used for the less tangible. If you fail to pay the rent, your landlord might reciprocate by evicting you. If your crush acts like he doesn't know you exist when you pass him in the halls, it's probably safe to say he does not reciprocate your amorous feelings.