THese were the months as celebrated by the Athenians, and the approximate time they were held (subject to lunar positioning)
BOEDROMION (Sept.- Oct.): literally means "to run shouting". "Boedromia" were games "in memory of the succor given by Theseus against the Amazons".
PYANEPSION (Oct.- Nov.): Derived from "puanon hepsein", i.e. "of a dish of beans cooked and offered". Puanepsia was an Athenian festival during such month in honour of Apollo..
MAIMAKTERION (Nov.- Dec.) : From "Maimaktes" an epithet Zeus at Athens meaning "boisterous, stormy"
POSEIDAION (Dec.- Jan.): : From "Poseidaia", games in honour of Poseidon.
GAMELION (Jan.- Feb.): : Possibly from "gamela", "offerings made on the occasion of a marriage."
ANTHESTERION (Feb.- Mar.): : Month during which Anthesteria, a three-day "Feast of Flowers" in honor or Dionysos was celebrated
ELAPHEBOLION (Mar.- Apr.): : Month during which the Elaphebolia festival of Artemis was celebrated . Elaphebolia, with all due respect to the PC crowd, means, "shooting of deer"
MOUNYCHION (Apr.- MAy.): Named after Mounuchia, a harbour in Piraeus. Munychia was also an epithet Artemis, who was worshipped there. Regrettably, the harbor of Mounuchia has been renamed today because "mouni" in contemporary Greek means vagina. It is presently a tourist trap with lotsa restaurants selling seafood.
THARGELION (May.- Jun.): Named after a festival a festival of Apollo and Artemis held at Athens during such month. The word is possibly etymologically related to "thalpo", meaning "heat, soften by heat".
SKIROFORION (Jun.- Jul.): Month during which the festival of Skiroforia or Skira, festival of Athena celebrated by Athenian women. I am not quite sure of the etymology, "skira" means " hard land overgrown with bushes" and "fero" "To wear, to carry".
HEKATOMBAION (July): During such month, oxen were offered to the Gods. Literally means " an offering of a hundred oxen"
METAGEITNION (August): Month during which the festival of Metageitnia was held: this celebrated "emigration" ("metoikismos"). From " meta and geiton", i.e. people "changing their neighbours".
During leap years, the month Poseidaion B' was added.
Source: Mostly from the Liddell&Scott "Lexicon of Classical Greek"