Its different based on preferred playstyle. For me, these are the stages
Early game - Starts at the beginning and ends when you can build destroyers. This is because with destroyers you can engage in offensive wars as you can finally fight spaceports without getting decimated. This part of the game is mostly exploration and planning expansion routes.
Early mid game - Starts when you can build destroyers and ends when you can build cruisers. Destroyers allow you to effectively wage wars of aggression. Cruisers are your first "capital" ships, which means less death stacks and more combined arms. Mineral spending decreases as you run out of stuff to build, and energy per month costs soar. Rivalries are declared.
Early late game - Starts when you can build cruisers and ends when you can build battleships. Battleships complete the combined arms part of combat (close range + long range combos etc.) allowing for huge wars. Minerals are always in surplus, and energy per month teeter on going negative unless you have a lot of trade deals. Wars of expansion against empires in your way are declared. Federations are formed, and often get in your way.
Late game - Starts when you can build battleships. Empires are huge and bloated now, with multiple fleets. Smaller empires are irrelevant, and often vassals. Only a few, obvious powerhouses remain. You should concentrate on your victory conditions now. Post-Banks, this would be the stage you focus on gene-modding (for genetic evolution ascension perk), cybernetic implants (synthetic evolution ascension perk) or psionics (psionics evolution ascension perk). Federations, militarist empires, FE's/AFE's and end game crisis's are the problems you face in this part of the game. Wars are often finals ones, to ensure your dominance of the galaxy.
Endgame - When you control ringworlds and have Titans. You should be the strongest power in the galaxy, and all that remains is to finish the job. FE's/AFE's should all be defeated, crisis's should be neutralized and other empires should be much weaker than yours. All thats left is cleanup, then the victory screen.