Investigators might look for several things while examining brass found at a crime scene.
The first is a human fingerprint left on the brass if the shooter was careless enough to load his weapon with bare hands.
Powder residue inside the case would be tested to determine what powder was used and if the cartridge was reloaded or factory ammo.
The extractor and ejector marks would be examined to try and determine what type of firearm was used. The configuration of these marks would be checked against a data base for this determination.
The extractor, ejector, and firing pin marks would be microscopically compared to cases found at other crime scenes to determine if the same weapon was used in all of the crimes. The machined surfaces of the extractor, ejector, and firing pin, leave a unique imprint (or fingerprint) where they contact to brass. Rough surfaces in the firing chamber can also leave a unique imprint on the outside of the brass case.
If a suspect is found and his weapon is recovered, test cartridges would be fired and the extractor, ejector, and firing pin marks would be microscopically compared to cases found at the crime scene to help tie the suspect to the crime.