Martius (31 days)
Aprilis (30 days)
Maius (31 days)
Iunius (30 days)
Quintilis  (31 days)
Sextilis (30 days)
September (30 days)
October (31 days)
November (30 days)
December (30 days
July was Quintilis renamed to honor Julius Caesar. His grandnephew was Augustus who defeated Marc Antony and Cleopatra, and became emperor of Rome.
To honor him, the Roman Senate chose the existing month Sextillus (sex means six). Why that month? In the Senate's decree:
Whereas the Emperor Augustus Caesar, in the month of Sextillis . . . thrice entered the city in triumph . . . and in the same month Egypt was brought under the authority of the Roman people, and in the same month an end was put to the civil wars; and whereas for these reasons the said month is, and has been, most fortunate to this empire, it is hereby decreed by the senate that the said month shall be called Augustus.
But there were a few problems with the choice. Julius's month, July, had 31 days, and Sextillis only had 30. That seemed like a slight to Augustus.
The Julian calendar had a nice alternation of months with 30 and 31 days (with the exception of February with 29).
The Senate took a day from February and reduced it to 28 days (29 in a leap year).
Of course, adding that extra day to August meant that July, August, and September would all have 31 days. Three long months in a row didn't sit well with them. So, they also switched the lengths of the last four months, and so we got 30 days in September, April, June, and November.
Julius and Augustus were the only Roman ruler to ultimately end up with months named after them. Emperor Claudius renamed May for himself and Nero instituted Neronius for April, but neither had any holding power and were reverted.