What is a 3 speed flash-o-matic automatic?

A Borg Warner transmission Rambler used in the early 60"s. It had a push button gear selector mounted on the dash to the left of the steering column.

Not all Flash-O-Matics were push-button shift, only 1958-62 "big" Ramblers used the push-button shifter. AMC (Rambler) started using Borg-Warner automatics (calling them "Flash-O-Matic") in 1957, and continued using them through 1970. The Flash-O-Matic name went away after 1966, with "Shift-Command" taking over. Shift-Command trannys used the now standard P-R-N-D-2-1 shift pattern. Flash-O-Matic used P-R-N-D2-D1-L. D2 started in second gear for better economy on level or near level surfaces and light loads, and also made it easier to start under slick conditions (like in snow and ice). D1 starts in first gear. Both D positions automatically shifted up through the gears. L was first only and didn't automatically shift. There were several versions -- M-8, M-11, M-12 were HD with cast iron case (and the only models through 62, used behind sixes and eights), M-35,36,37,40,42,44 were aluminum cased light to medium duty versions used behind sixes and small V-8s.

I own a 62 amc rambler classic could I use a 390 with the stock transmission and stock motor mounts

No. The 62 Classic still used four point motor mounts -- two in front of the motor, two on the bell housing. The crossmember in the 62 is strong enough to carry an engine though. You can weld or bolt mounts to the crossmember for the motor, but will have to make a transmission crossmember or use a universal type. You will also need to use a different rear axle and build a suspension. Universal four link kit or something like that. You can keep the original springs and modify the original panhard bar mounts to fit another axle. Late Ford Ranger axles are a good fit, and 4.0L V-6 Rangers use an 8.8" axle. Explorer axles are too wide. Use the original axle spring seats and bolt to the top of the new axle, use the original rear springs. None of this is too hard for anyone who has done some fabrication, but it's not bolt-in. You might want to join the AMC Forum (www.theamcforum.com) or Hotrodders.com to get more detailed answers.