The noun galaxy is from post-classical Latin galaxias, denoting the Milky Way, from Hellenistic Greek γαλαξίας (= galaxias), short for γαλαξίας κύκλος (= galaxias kuklos), milky circle, from ancient Greek γάλα/γαλακτ- (= gala/galakt-), milk.
Originally therefore, galaxy, often with the and capital initial, denoted the Milky Way, that is, the diffuse band of light stretching across the night sky that consists of millions of faint stars, nebulae, etc. In The House of Fame, the English poet Geoffrey Chaucer (circa 1342-1400) wrote:
Se yonder, loo, the Galaxie,
Which men clepeth the Milky Wey
For hit ys whit.
See yonder, lo, the Galaxy,
Which they call the Milky Way
Because it is white.
In later use, galaxy came to denote a feature, system, etc., resembling the Milky Way, hence any of the numerous systems, often of millions or billions of stars, held together by gravitation and containing other matter such as gas and dust, which exist throughout space as distinct bodies.
Among these galaxies is the Milky Way galaxy, that is, the disc-shaped star system that contains the solar system and the visible stars and whose plane of maximum star density gives rise to the observed Milky Way. The Milky Way system is a disc-shaped spiral approximately 100,000 light years across, containing at least 200 billion stars. The sun is located about two thirds of the way outwards from the centre.
The Greek γάλα/γαλακτ- (= gala/galakt-), milk, is the base of the adjective galactic, which has two main meanings: (1) of, or relating to, a galaxy, and (2) lactic, as in galactic acid, synonym of lactic acid (the Greek word is probably related to Latin lac/lact-, milk.)
This is why, surprisingly enough, there is an etymological connection between the words galaxy and lettuce. The latter is ultimately from classical Latin lactuca, lettuce, from lac/lact- and the suffix -uca forming nouns; it was a reference to the milky juice of the plant.
The ending of lettuce is apparently due to Anglo-Norman letuse, inferred singular of letues, plural of letue (the modern-French word is laitue). Similarly, quince was originally the plural of the obsolete quine, meaning quince; probably via use as a collective singular, quince came to denote a single fruit.