Short answers, pulled from the story.
Julius Caesar reformed the Roman calendar in 46 BC. To reset the calendar to its correct solar position, the year 46 BC was extended to 446 days by inserting three intercalary months. The resulting Julian calendar was a purely solar system of 365 days with a leap day every fourth year.
The Kalends, Nones, and Ides were the three fixed anchor points within each Roman month. The Kalends fell on the first day of every month; the Nones on the seventh day of full months and the fifth of shorter ones; the Ides on the fifteenth of full months and the thirteenth of shorter ones. Romans counted days backwards toward these points rather than forwards from the start of the month.
The Republican calendar had only 355 days, creating a growing gap from the 365.25-day solar year. Every other year an intercalary month called Mercedonius was inserted between the 23rd and the 24th of February to restore alignment. The Pontifex Maximus controlled when this insertion occurred, which was frequently exploited for political purposes.
The nundinal cycle was Rome's eight-day market week. Market days, the nundinae, fell at the end of each eight-day period, though Romans counted them as nine days using their inclusive counting method. The cycle was shared with the Etruscans and was displaced by the seven-day week after Constantine made Sunday an official day of rest in AD 321.
January and March were named for the gods Janus and Mars; July and August honored Julius Caesar and the emperor Augustus. September through December are Latin ordinal numbers for seventh through tenth, reflecting their positions when the year began in March. February may derive from the Februa purification festival, and the origins of April, May, and June remain uncertain.
Astronomical events recorded by Livy show the Roman civil calendar was four months out of alignment with the solar year in 190 BC and still two months off in 168 BC. By the time Caesar acted in 46 BC, the calendar was displaced by enough that an entire season had shifted.