Learn German: Ich wohne bei meiner Familie.

In German, bei is used when you mean you live at the home of someone or are staying at their place. If you said Ich wohne mit meiner Familie, it just means you are living together with them (like sharing an apartment), but bei specifically gives the feeling that you are staying at their house.

Why is the word for "my" spelled meiner here instead of just meine?

The preposition bei always forces the next phrase into the Dative case. Because the word for family is feminine (die Familie), the word for "my" (meine) changes its ending to -r in the Dative case, becoming meiner.

Are there other words like bei that always force this Dative change?

Yes! bei belongs to a group of prepositions that strictly require the Dative case. Other common ones you are learning right now include mit (with), von (from), and zu (to). Whenever you use these, the words that follow them must take Dative endings.

Does bei always translate to "with" in English?