The Romans had names that described points in the month. Ostensibly these names came into existence during the Roman lunar calendar. The Ides of a lunar month would be the full moon. Although we believe these terms to originate on their lunar calendar, we have no record of a Roman lunar calendar.
The Romans followed traditions even when the purpose of that tradition no longer existed. Hence, when they transitioned to a solar calendar, they still used their lunar terminology. Furthermore, that terminology fit with their monthly count. They didn’t count days sequentially like we do today. Rather, they would count the days to the next point in the month (or even the next month).
The Ides, being the full moon on their lunar calendar, would be the middle of the month on their solar calendar. Ergo…the 15th. Every month has an Ides. We probably remember the Ides of March as that was the day on which Caesar was assassinated, and was made famous in Shakespeare’s soothsayer’s comment to Caesar to “Beware the Ides of March”. To which Caesar latter replies “The ides of March are come” (meaning it is now the 15th of March, and I’m still here). To which the soothsayer responds…”Ay, Caesar; but not gone” (which means…the 15th has yet to end)… And we all know what happens next.”
All the other months have Ides. It’s just that none have been so eloquently anchored in the English psyche than the Ides of March, and its historical significance of a warning of dangers ahead.