Technically, all the rides which are under the ACP or BRM umbrella are randonnees. However, I (and others) tend to use the word randonnee for rides of 1000K and more - usually under the BRM umbrella
Technically, a brevet is the card we get signed. Brevet is the french word for patent, or I guess, proof that you've completed a randonnee. However, I (and others) tend to use the word brevet for rides of 600K and less - usually under the ACP umbrella.
Audax is a specific style of randonneuring where all the cyclists ride together. These events are actually quite rare these days, but in the early days of randonneuring, there was one group who always rode this way, and one group who rode brevets/randonnees the way most are run today. I believe the audax group merged with the randonneuring group back 30 or 40 years ago.
Fleche means arrow because there are supposed to be several teams, each starting in a different location, but each ending in the same "bull's eye" (so to speak). Each team does sort of ride the event audax-style.
Permanents are much more open ... you can ride them wherever and even design your own route ... but you do have to have it all approved.