After Krypton exploded and freed the Black Zero from the Phantom Zone, General Zod commandeered the starship and searched the old Kryptonian outposts scattered across the universe for survivors. Finding no survivors, they scavenged what they could fit on their ship - armor, equipment, a World Engine, and 2 dropships.
After discovering Kal-El and the Growth Codex on Earth, General Zod positioned Black Zero in Earth's orbit, and sent the 2 dropships down, lead by Faora-Ul to retrieve Kal-El and the human Lois Lane, before returning them to the Black Zero.
After searching their memories on the Black Zero for the Codex's location, General Zod led 2 dropships down to Smallville, landing on the Kent Farm. Kal-El, however, escaped the Black Zero and attacked Zod. A dropship, piloted by Tor-An, followed and attacked Kal-El, blasting him away away from the weakened Zod, and took him back to Black Zero, before returning for Faora-Ul and Nam-Ek, shooting down several human military planes in the process.
General Zod later personally flew a dropship to the location of the Scout Ship in the Arctic, to secure the Genesis Chamber, and left it there on the mountain, as he flew the Scout Ship away to Metropolis.
The United States Army discovered the abandoned dropship, and moved it back to the United States Department of Defense, where they could properly examine and try to reverse engineer its highly advanced technology and capabilities, with help from LexCorp and S.T.A.R. Labs. They confiscated all the weapons and other equipment aboard as well for examination.
A Kryptonian Dropship is a large, black starcraft that vaguely resembles a large round beetle in appearance. It is capable of impressive maneuverability, and great speed of at least 196 m/s, thanks to its two powerful jet thrusters. Given that human passengers on board the ship are able to not only survive, but stand up and move around during such lethal acceleration, it appears that the ship is also equipped with some kind of inertial dampening technology to protect the crew and systems. For weaponry, the Dropship possesses a single cannon underneath the cockpit that emits thick blue plasma bolts that are capable of bringing down a A-10 Thunderbolt II with a single hit. Indeed, such a shot was powerful enough to send Kal-El flying into a truck, with enough force to greatly dent it and cause him burning pain in the process.