Learn Swedish: Vårt hus är mycket gammalt.

The word for "house" (hus) is an ett-word (ett hus). Possessive pronouns must match the noun they belong to. Because it's an ett-word, "our" becomes vårt. If it were an en-word, you would use vår, and if it were plural, you would use våra.

Why is there a -t at the end of gammalt?

Adjectives also have to match the gender of the noun they describe! Because hus is an ett-word, the adjective for "old" changes from its basic form (gammal) to the ett-form, which adds a -t to become gammalt.

I thought mycket meant "much" or "a lot". Can it also mean "very"?

Exactly! mycket can mean "much" (as in mycket kaffe for "a lot of coffee"). However, when you put it right in front of an adjective like gammalt, it translates to "very".

Do I need to put hus in the definite ("the") form, like huset, since it's a specific house?