Learn Russian: Наша чёрная собака дома.

They are describing собака (dog), which is a feminine noun. In Russian, adjectives and possessive words must match the gender of the noun they belong to. So we use the feminine forms: наша instead of the masculine наш, and the standard feminine adjective ending -ая to make чёрная.

If I wanted to say "our black cat" (кот, masculine), how would the adjectives change?

Because кот is masculine, you would need to change the possessive word and the adjective to their masculine forms. The sentence would become: Наш чёрный кот дома.

Where is the word for "is" in this sentence?

Russian almost never uses the present tense of the verb "to be" (am, is, are). You just put the subject and its location together, so the sentence literally translates as "Our black dog at home."

Why is the word for "at home" дома and not just дом?

Дом is a noun that means the physical building (a house). Дома is a specific word used to mean the location "at home." If you said Наша чёрная собака — дом, it would actually mean "Our black dog is a house!"