Learn Farsi: عید امسال در خانه‌ی فامیل بودیم. - This year at the festival we were at our relatives' house.

This is the ezafe connector, which links two words together to show possession, like saying house of relatives. Because خانه ends in a silent h, we cannot just add a short e vowel sound. Instead, we must add a ye sound to bridge the words, which is written with the letter ی. So it becomes خانه‌ی فامیل.

Is فامیل used exactly like the English word family?

Not quite! In English, family usually refers to your immediate family. In Farsi, فامیل generally refers to your extended family or relatives like aunts, uncles, and cousins. If you want to talk about your immediate family, you would use the word خانواده.

How are the words عید and امسال connected?

They are connected by a hidden ezafe, which is a short e sound, making it sound like عیدِ امسال. This literally means the festival of this year. Since this short vowel is not usually written in the Persian alphabet, you just have to remember to pronounce it to link the noun and its time modifier.

The translation says our relatives' house, but I do not see the Farsi word for our in the sentence. Why is that?

You are right, there is no word for our, which would be مان, attached here. The Farsi sentence literally says in the house of the relatives. When the context is perfectly clear from the verb بودیم meaning we were, speakers often drop the possessive pronoun. It sounds much more natural to add our in the English translation.