Learn Italian: Questa è un'altra casa.

The word un'altra is a combination of the feminine article una (a/an) and the adjective altra (other). Because una ends with a vowel and altra starts with one, the final a of una is dropped and replaced by an apostrophe to make it smoother to say.

Would I also use an apostrophe for a masculine noun, like un altro?

No, and this is a very common tricky spot for beginners! For masculine words, the indefinite article is simply un. Because un does not end in a vowel, there is no letter to drop. You only use the apostrophe for feminine words like un'altra, while masculine combinations are written as two separate words without an apostrophe, like un altro.

Why do we start with Questa instead of Questo?

Questa is a demonstrative pronoun meaning "this", and it always has to match the gender of the noun it refers to. Since casa (house) is a feminine noun, we must use the feminine form Questa. If we were talking about a masculine noun, we would use Questo.