A felicitation is when you congratulate someone. Your best friend might express his felicitation on your birthday by giving you flowers.
You'll almost always see the noun felicitation in its plural form, felicitations. In fact, when your neighbors announce their upcoming marriage, you can say, "Felicitations!" just as you might say, "Congratulations!" Either way, you're expressing your good wishes. Felicitation comes from the verb felicitate, which means "congratulate," and has a Latin root, felicitas, "happiness," from felix, "happy or fortunate."