It all starts with a basic concept which is then fleshed out:
Dig Dug - move underground by digging through dirt
Donkey Kong - rescue the princess by ascending a structure
Centipede - shoot at an enemy as it slowly approches you
Defender - fly a spaceship over a planet shooting enemies
Missile Command - shoot down approaching missiles
Galaga - shoot enemies which move in patterns
As you can tell, the basic gameplay is all based on relatable concepts which have been seen before. What makes these games special are the additional elements:
Dig Dug - enemies take time to kill (by being inflated) and can float through walls. You could argue that the digging concept was brand new, which may be right.
Donkey Kong - barrels roll along your path and randomly use shortcuts you can take (ladders) and also come from behind you
Centipede - enemy movement pattern changes based on environment, which is randomly and dynamically created
Defender - defend civilians in addition to attacking enemies, advanced effects for its time
Missile Command - you have advanced control over your projectiles and damage is dealt indirectly, forcing you to think ahead
Galaga (is actually a successor to Galaxian) - clever/fun enemy patterns, ability to snatch player
If you are curious about new and experimental gameplay, look into Jonathan Blow's prototypes. He also has some nice YouTube talk videos you should check out, like
.