Galaga is a spiritual successor to Galaxian. Building on the formula established by Galaxian, Galaga introduces several new concepts to the fixed vertical shooting genre that made it incredibly popular in the arcades. It can still be found in arcades to this day, and performs rather well next to more modern day offerings.
The most notable new feature of Galaga is the chance to increase your fire power by intentionally allowing one of the Boss Galagas to capture your Fighter, and rescue it with your next Fighter, returning the original Fighter to your side. Upon rescuing your ship, you are granted side-by-side double fire power, albeit with an increase in size, and therefore, vulnerability. Another popular feature of Galaga was its presentation of Bonus Rounds (dubbed "Challenging Stages") between the action which offered large point awards for perfection.
Galaga seems to represent something of a "sweet spot" in game design, having performed better than either its predecessor or its successor, Gaplus. However, despite its incredible popularity in the arcade, it saw comparatively few official home conversions in the United States. Several homebrew programmers supplied alternate versions stateside, while Japan saw an incredible number of conversions for popular computer models only available in the land of the rising sun.
There is no official story to Galaga. You are simply the only pilot capable of staving off the invasion of the Galaga army, a collection of giant sized insect looking aliens.