Affinity was introduced with the Mirrodin block as a main mechanic for the block which was Artifact themed. It is also the name of a very powerful deck which utilizes this ability to the greatest extent. During it's peak the DCI banned multiple cards in an effort to curb the deck and return it to a manageable power level.

702.38. Affinity

• 702.38a. Affinity is a static ability that functions while the spell is on the stack. "Affinity for [text]" means "This spell costs you {1} less to play for each [text] you control."
• 702.38b. The affinity ability reduces only generic mana costs; it doesn't reduce how much colored mana you have to pay for a spell.
• 702.38c. If a spell has multiple instances of affinity, each of them applies.

Rules Translation

• 702.38a. - This is an always on ability, it is not triggered, it is constantly modifying and changing any instances where it is applied.
• 702.38b. - This only decreases the cost by colorless mana, it does not remove any colored mana from the cost.
• 702.38c. - Affinity stacks so if you have a spell that somehow has Affinity 1 + Affinity 1 it actually costs 2 less to play.