The Romans had special names for 3 specific days in each month. The system was originally based on phases of the Moon (Luna), and these days were probably declared when the lunar conditions were right. After the reforms of Numa Pompilius, they occurred on fixed days.
Kalends - first day of the month, from which the word "calendar" is derived. Interest on debt was due on Kalends.
Nones – depending on the month, could be the 5th or the 7th day; traditionally the day of the Half Moon
Ides – depending on the month, could be the 13th and 15th day; traditionally the day of the Full Moon
Days were numbered in a way that is quite different from the modern Western calendar. The Romans did not count the days of the month retrospectively, looking back to the first of the month (that is: 1st, 2nd day since the start of the month, 3rd day since the start of the month). They counted forward to their named days. Also, to the distress of moderns trying to work out dates in Roman calendar documents, they counted inclusively, so that September 2 is considered 4 days before September 5, rather than 3 days before.
The example of September
The following example spells out how days were named for the pre-Julian September, which had only 29 days. It shows the Roman form of the date, the translation, and how we would say it today. The Romans used abbreviations: "a.d." = "ante diem" = "day before", "prid." = "pridie" = "the day before", "Kal" = "Kalends" etc.
Kal. Sept. = Kalends of September = September 1
a.d. IV Non. Sept. = 4 days before the Nones of September = September 2
a.d. III Non. Sept. = 3 days before the Nones of September = September 3
prid. Non. Sept. = the day before the Nones of September = September 4
Non. Sept. = Nones of September = September 5
a.d. VIII Id. Sept. = 8 days before the Ides of September = September 6
a.d. VII Id. Sept. = 7 days before the Ides of September = September 7 and so on till
a.d. III Id. Sept. = 3 days before the Ides of September = September 11
prid. Id. Sept. = the day before the Ides of September = September 12
Id. Sept. = Ides of September = September 13
a.d. XVII Kal. Oct. = 17 days before the Kalends of October = September 14
a.d. XVI Kal. Oct. = 16 days before the Kalends of October = September 15 and so on till
a.d. III Kal. Oct. = 3 days before the Kalends of October = September 28
prid. Kal. Oct. = the day before the Kalends of October = September 29
Kal. Oct. = Kalends of October = October 1
Notice that by counting inclusively and by having a special name for the day before a named day the Roman calendar loses the possibility of saying: 2 days before a named day. Also, after the Ides, the date no longer mentions September, but is counting down towards October.
When Julius Caesar added a day to September, he didn't add it to the end of the month. Rather, the new day that got added was the day after the Ides:
a.d. XVIII Kal. Oct. = 18 days before the Kalends of October = September 14
As a result, the position of all the following dates in September got bumped up by one day. This has some unexpected effects for modern readers. For example, the emperor Augustus was born on 23 September 63 BC. In the pre-Julian calendar this is 8 days before the Kalends of October (or, in Roman style, a.d. VIII Kal. Oct.), but in the Julian calendar it is 9 days: a.d. IX Kal. Oct. Because of this change, in some parts of the Empire his birthday was celebrated on both dates, i.e. (for us) on 23 and 24 September.
Beware the Ides of March
But continue to ignore the Nones of July. Wikipedia has more information about the Roman calendar than you ever cared to know.