From Sonic Retro
The Death Egg is a massive orbital fortress, constructed by Dr. Ivo "Eggman" Robotnik. A true study in evil mega-engineering, the spherical spacestation resembles the face of Eggman himself, with concave eye-sockets and a moustache rendered in kilometres of vacuum-hardened plasmasteel. Hosting some of the Doctor's most deadly creations of all time, this vast stronghold serves as the scientist's final base in the early Sega Genesis-era games. The Death Egg is a tribute to the Death Star, a space station in the Star Wars series.
Death Egg Saga
The station's first appearence saw it functioned as the driving force for a three-game meta-plot, starting with Sonic the Hedgehog 2. Death Egg fulfilled the eponymous Death Egg cliché as the final level. From the level select screen, we can see that the Death Egg is roughly spherical even in its original incarnation. Death Egg Zone contained only the final two bosses, Mecha Sonic (aka Silver Sonic) and Dr. Robotnik's Giant Mech. After Sonic defeated the Doctor, the Death Egg exploded - but it wasn't entirely destroyed, and the damaged space-station survived an atmospheric re-entry to crash-land on Angel Island.
In Sonic 3, Robotnik repaired the Death Egg and was about to re-launch when Sonic arrived in the Launch Base Zone. Sonic quickly boarded the giant space station and defeated Eggman again in a battle on the underside of the ascending station's hull.
In Sonic & Knuckles (or Sonic 3 & Knuckles) the Death Egg falls out of the sky and lands in the volcano of Angel Island (otherwise known as the Floating Island) where Lava Reef Zone and Hidden Palace Zone were situated. In the end of the game, the Death Egg was launched again, using the power of the stolen Master Emerald. It then became a two-Act Zone playable by Sonic and/or Tails containing an Act 1 boss and three Act 2 bosses (including Tails' final boss). After defeating the final boss, the Death Egg was (presumably) destroyed.
Throughout the storyline, it is never explained what the Death Egg does or is supposed to do.
In the arcade game Sonic Championship (Sonic the Fighters in Japan), Robotnik went to make the Death Egg II, which once again was home to the 2 last bosses. Although retaining the shape of the original death egg, it was different in the fact many pronged girders came off in different directions - these had no obvious purpose. It was then destroyed and never heard of again.
Later on, in Sonic Battle, Robotnik rebuilt/reused the Death Egg for use of his "Final Egg Blaster". The final battle against Emerl was situated there. The game didn't tell what happened to it after Emerl was defeated by Sonic, but it was presumably destroyed. Seen from Holy Summit, the Battle Death Egg is hemispherical and looks a more like the Space Colony ARK.
The Dead Line Zone in Sonic Rush is similar to the Death Egg.