From 1920 to 1947 the US Army Air Service/US Army Air Corps/US Army Air Force used a designation system based on mission category, with each model in a category numbered in turn.
From 1922 to 1962 the US Navy (including US Marine Corps) used a completely separate designation system, based on both mission and manufacturer.
From 1947 to 1962 the US Air Force used a mission-based system different to, but derived from, the 1920-1947 system. For instance, the Lockheed P-80 `Shooting Star' (Pursuit) was re-designated as F-80 (Fighter), but the Convair B-36 bomber kept its previous designation.
From 1962 to now a joint system of mission-based designations was used, with most of these re-starting from 1. Various previously-designated models (such as the F-111) were not re-designated, however.
A comprehensive explanation can be found at [1] (as at 2002-08-21).
Intention is to split into 3/4 sets of lists, headed by explanatory table, one for each of US Army, US Navy, and US Air Force. Plus maybe another for WW1 aircraft.