A salt is a compound composed of two charged groups. One positively charged, one negatively charged. For drugs such as adderall, one is a positively charged amphetamine (with an extra hydrogen as a proton) and the other half is a negatively charged counter ion (either sulfate, sacharate or aspartate). The reason it is given as a salt, is that charged molecules dissolve more easily in water. If it was not charged, it would not dissolve and be less effective when taken orally.

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In the case of lithium, it is usually lithium cations (positively charged) with carbonate, bromide or other counter anions. The truth is, we don't really know exactly how lithium works in the brain. It interacts with a number of things but we can't pin down which affects particularly help with mood stabilization.