Romans counted forward to the next named day, inclusively. The named days are the Kalends (1st), the Nones (5th, or 7th in Mar/May/Jul/Oct) and the Ides (13th, or 15th in those four months). The Ides of March is a.d. XV Kalendas Apriles in nobody’s joke — it is just the 15th.
Inclusive counting: the day itself is 1. Pridie = the day before. Ante diem III = two days before in modern exclusive counting. Month names in the accusative after Kalendas / Nonas / Idus. Leap-year extra day is a.d. bis VI Kal. Mart. — this desk only flags the year type, it does not rewrite the whole bissextile dossier.