1 Ability Power (AP) = 1% increase for all magic damage, true damage, shielding and healing. Almost all spells deal magic damage except for Volibear's, Draven's, Rengar's, Graves's, and partially Lucian's. Their spells modify the physical damage of basic attacks. This is from @RiotAugust's tweet.
Items, team synergies, and spells in TFT can increase a champion's AP. AP increases the effects of spells but it's unclear exactly how it works. Riot August, a game designer on Riot's TFT team, clarified what AP does on Twitter. One important thing to note is that since patch 9.14, AP no longer modifies the effects of items (e.g. the shield from the Locket Of The Iron Solari doesn't increase in size when you increase a champion's AP).