A month is a portion of a year corresponding roughly to the passage of the earth round the moon (a 'lunar month', of which there are, in fact, thirteen). There are twelve terrestrial months in the Gregorian Calendar. They are named after pagan gods, goddesses, festivals, and events (January (from Janus), February (festival for forgiveness), March (Mars (god)), April (possibly from aperire, to open, as in buds and blossoms), May (Maia), June (Juno)), Roman emperors (July, August) and numbers (September to December inclusive).
The ninth to twelfth months are named after the Latin words for seven to ten because in Roman times the year started with March. February was chosen to have the extra day added every four years because it was the last month of the year.
The number of days per month varies, but a handy way to remember which is which is with the following rhyme:
Thirty days hath September
April, June, and November
All the rest have thirty-one
Except for February alone
Which has twenty-eight days clear
And twenty-nine in each leap year.