The story
Imieniny is Polish for a name day, the feast of the saint whose name you carry. It is built from imię, name, and like many Polish words for celebrations it only exists in the plural. The same word covers the day itself and the party that goes with it.
Name days grew out of the Christian calendar of saints, and the church long encouraged them over birthdays, which had a pagan reputation. In parts of Poland imieniny have traditionally mattered more than birthdays. Friends and family gather at the celebrant's home, and flowers and small presents turn up at work as well. Local calendars print the names for each day, so there is little excuse for forgetting a colleague's.
Younger Poles increasingly make more of their birthdays, but the habit is far from gone. Russia has the same institution under a related word, imeniny, and before 1917 it was often thought at least as important as a birthday.
English does have name day, and some Catholic and Orthodox families keep one. What it does not have is the social machinery. In Poland the name day comes with a date everyone can look up, a reason to bring cake to the office and a fair chance that three people on the same floor are celebrating at once.
On a Polish calendar in late July, for instance, a single date can list several names, and every one of those people expects at least a phone call.