What was the forensic process underlying the recent news regarding MH370 debris?
If you bring an NTSB investigator a piece of 2024 sheet aluminum with zinc chromate primer and a urethane topcoat all they can tell you is "It's 2024 sheet aluminum, with some paint on it." - BUT if you bring them a piece of 2024 aluminum riveted to a  structure that looks like this:

and tell them "I found it on an island in the Indian Ocean" they can infer  lot more.



In this particular case they've found some debris, and they know a few things:


The debris was found in an area where it's reasonable it could be part of MH370.
The location isn't conclusive in and of itself, but it's certainly a factor that strongly implies this could be part of the missing aircraft.
The part LOOKS like it came off an airplane.
It's aircraft aluminum, riveted to a skeleton that gives it an airfoil shape. That warrants further investigation by itself (even if we didn't lose an airplane in that area recently it's clear that some part fell off some airplane at some point in time, and investigators generally want to know which airplane had bits falling off).


That leaves us with the question Could this part have come from the airplane we lost?  which is the stage we're at now, and there are many ways to make this determination.


You might get lucky and recover the aircraft ID plate, or a chunk of the tail with the N number.
There's confirmatory testing to be sure it's not a well-perpetrated hoax, but these are pretty definitive identification elements. Unfortunately that's not the case here, so we move on.
If you recover what appears to be a control surface you'll usually find a manufacturer stamp somewhere on the part, or a serial number plate. They'll certainly be inspecting this debris to look for that kind of data tag. (The data tag below is from a Piper control surface but Boeing probably uses similar plates).
  
If you don't find a data tag maybe there's airline livery on the debris.
This at least lets you know whose plane you've found, but that's not applicable here as the debris in question appears to be painted basic white.
Failing all of that you compare what you've found to manufacturer's drawings of the aircraft you think it came from and see if the dimensions match any parts or sections of that aircraft.  


There are other possible steps and confirmatory testing involved (metallurgy, mass spectrometry on the paint, etc.) to determine if it's likely this part came from the aircraft you think it did, but that's getting into the territory where you'd be better off signing up for training courses in accident investigation.