The Roman names for the months are familiar because they are also used, with small changes, in English and most other European languages. In Latin the words were actually adjectives attached to the noun mensis (month) but the noun was often omitted both in speech an in writing.
Iānuārius Aprīlis Iūlius (Quīnctīlis) Octōber Februārius Māius Augustus (Sextīlis) November Martius Iūnius September December
Months in –us have endings like bonus, those in –er follow the pattern of ācer,ācris,ācre and Aprīlis is like omnis, omne. The last six months got their names by counting from the start of the year, which originally began on 1st March (so September is `month seven’, not `month nine’). New Year’s Day was moved to 1st January in the 2nd. century B.C. so that the consuls (the chief Roman government officials, who came into office at the start of the year) would have time to get from Italy to Spain before the weather became suitable for military operations. The seventh and eighth months were re-named in honour of Julius Caesar (100-44 B.C.) and of Augustus (63 B.C. – 14 A.D.), the first Roman emperor. Iānuārius and Martius were named for the gods Janus and Mars respectively and similarly Iūnius probably derives from the name Juno. Aprīlis may be from Greek Aphrodite (Venus) via the Etruscan form Apru, or alternatively, in view of its original status as the second month of the year, it derives from a hypothetical early Latin word *ap(e)rilis "the following, the next" , which itself may come from Proto-Indo-European *apo- "away, off." Māius probably comes from the earth goddess Māia, whose name perhaps derives from PIE *mag-ya "she who is great". Februārius was named for the Februa (`Purification') festival which was held on its 15th day. See www.etymonline.com for more information, including the Old English names of the months which the Latin ones replaced.
The Romans had special, feminine plural names for thee days of the month: Kalendae -ārum (Kalends) 1st., Nōnae, - ārum (Nones) 7th of March, July, October and May, 5th of other months Īdūs, -uum (Ides) 15th of March, July, October and May, 13th of other months
The ablative case of these nouns was used to give the date of an event which happened on one of these special days: :
KalendīsIānuāriīs Nōnīs Iūliīs Īdibus Septembribus Īdibus Octōbribus on 1 Januaryon 7 July on 13 September on 15 October
Dates for the days immediately before the special dates were given with the phrase prīdiē and other dates in the month by counting back from these special days and using the phrase ante diem (`before the day’) with an ordinal number. The name of the special days was also in the direct object (accusative) case. The Romans reckoned inclusively, i.e. they counted both the day at the beginning and the day at the end of a period when working out its length. Thus the 11th. of March was five days before the 15th, not four.
prīdiē Kalendās Iūniās on 31 May (`on the day before the Kalends (1st) of June’)
ante diem quartam Īdūs Februāriās on 10 February (`on the fourth day before the Ides (13th) of February’)
ante diem tertiam Nōnās Iānuāriās on 3 January (`on the third day before the Nones (5th) of January)
Usually dates were written in abbreviated form:
Pr. Kal. Iūn a. d. IV Īd. Feb. a. d. III Nōn. Iān.
The website http://www.wilkiecollins.demon.co.uk/roman/calco1.htm provides a conversion table for Roman dates and the Cambridge Latin course site gives the current Roman date. If you want to date a letter in Latin, you normally put it at the end and add the verb dabam (I was giving (i.e. writing))..
There was an extra complication in the final days of February in leap years (introduced into the calendar by Julius Caesar in 45 B.C.): from the Ides (13th) of February onward every day up until the 24th (a. d. VI Kal. Mar.) was dated as if the month had only 28 days, but February 25th was called a. d. bis (twice) VI Kal. Mar. and the remaining days of the month calculated on a similar 29-day basis (see the table on the WilkieCollins website).
The Romans themselves originally used the names of the consuls ( the chief government officials, who normally served for just one year) to refer to a particular year. Thus `in 63 B.C.’ would be [M. Tulliō] Cicerōne et [C.Antōniō] Hybridā cōnsulibus, in the consulship of Cicero and Hybrida . Later on years were counted from the supposed date of the foundation of Rome in 753 B.C., using the phrase ab urbe conditā (from the foundation of the city), abbreviated a.u.c.:
annō septingentēnsimō quīnquāgēnsimō tertiō ab urbe conditā (annō DCCLIII a.u.c.) in the seven hundred and fifty-third year from the foundation of the city (in 1 B.C.)
annō septingentēnsimō quīnquāgēnsimō quartō ab urbe conditā (annō DCCLIV a.u.c.) in the seven hundred and fifty-fourth year from the foundation of the city (in 1 A.D.)
In medieval and also neo-Latin the year is normally given in the modern system and the same is often done for the day of the month. When giving historical dates, either annō Dominī or ante Christum nātum (before the birth of Christ) can be added if necessary.
diē vīcēnsimā prīmā Maiī mensis annō [Dominī] bismīllēnsimō sextō on the 21st day of the month of May in the year 2006 [A.D.] (on 21 May 2006)
Whatever system is used for the years, the Roman numeral has to be an ordinal and must be read as one compound number, not broken up as in English `nineteen ninety-nine’ etc. Note also that the `n' in the suffix ēnsimus, -a, -um for ordinal numbers is frequently omitted (e.g. vīcēsima for vīcēnsima).
The days of the days of the week (hebdomas, -adis (f.) or septimāna, -ae f) were named after the sun, moon and planets, most of which had taken their own names from those of Roman gods:
After the Roman empire became Christian, Sunday was known as diēs Dominica (`the Lord’s Day’), hence French dimanche, Italian dominica etc.
For general explanations of the Roman calendar and additional examples of date conversions, see also Latin via Ovid, pages 439-440 and So You Really Want to Learn Latin, Book II, pp.52-53. For the relationship between the Roman names for days of the week and the English ones, see the article `Why Wednesday?', published in the Guardian in January 2018.
 An alternative theory is that the change to January was made earlier than this and that it was the start of the consuls’ period in office which was moved to coincide with New Year’s Day in the 2nd. century. See the account of the calendar at: http://www.polysyllabic.com/?q=book/export/html/15
 You can remember which month have the later Nones and Ides by using this rhyme: In March, July, October, May The Nones are on the seventh day You can also remember the phrase `Beware the Ides of March’ (cavēĪdūs Martiās), said to Julius Caesar by a fortune-teller a few days before his assassination on 15 March 44 B.C.). The Nōnae (ninths’) were always nine days (or eight days by our way of counting) before the Ides.
 Nobody knows the exact year in which Christ was born. The sixth-century monk Dionysius Exiguus, inventor of our present system of numbering years, placed the birth at the end of 753 A.U.C. This is certainly wrong because Christ was born in the reign of Herod the Great, who died in 750 A.U.C. (4 B.C.)