A = Autoroute, aka Autobahn, aka Motorway. Motor vehicles only, two carriageways, usually 2 lanes in each direction, no traffic lights or other hindrances. Fast, boring (with some exceptions). And, apart from near big cities, they have tolls.
Speed limit 130 Km/hr, less when wet and some local restrictions.
N = Route Nationale. Long distance cross country road. May be single or dual carriageway. Reasonably fast (not as fast as Autoroutes), but sometimes goes through towns and villages. May have crossroads, traffic lights, roundabouts, pedestrian crossings, cyclists etc.
Speed limit 110 km/h if dual carriageway, 80 Km/h if single carriageway. Urban speed limit of 50 km/h
D = Route Départementale. The lowest level, could be any local road between villages.
Note the numbering is by Département. Each Département has a D1, D2 etc. Road numbers therefore change when you cross a Département boundary.
Same speed limits as Route Nationale