Part of Board exploded when Peltier Plate was wired t(o Board
You did a pretty good job of nuking your board. Gratz.
Basically what has happened: you connected 19V to the input of the 5V regulator (rated at 800mA max) and connected a 12V 50W device to the output.
I have no idea what a peltier will do when under powered, but at 12V it's going to want about 4A. That's 20x more than the 5V regulator can provide (also exacerbated by the massive amount of heat needed to be dissipated to drop 14V) - so it went pop.
In going pop it short circuited. That meant that 19V (or just short of) ended up going into the rest of your Arduino. The 3.3V regulator didn't like that (the bit that looks toasted), and neither will any of the rest of the components on the board.
In short: probably almost every component on the board has been nuked.
The moral of this story: don't connect high powered 12V devices direct to the Arduino. Things like peltier devices need their own dedicated power supply which can be controlled (as you would a motor or relay) through a power MOSFET.