AP does not boost auto-attacks at all, only AD does*.
*In most cases anyways, very few champs get a small amount of bonus damage on auto-attacks based on AP. Mainly due to their passives. (See Diana, Orianna or Varus for examples)
Depending on the ability, the damage primarily scales with either AP (Green) or AD (Orange). You can check this on each ability's tooltip. the (+) will be color coded as any of the color's I've listed. A small number of ability's can actually scale off of more than one source, and it will have two boxes [aka. (+)(+_) ] with each being a seperate color.
There are few other rare cases where they are based on other factors. One example is on max Mana (Blue, Ryze/Taric). Other's are based off your max HP (Red, Volibear/Sejuani) and Armor (Yellow, Malph/Taric)