"Mom is home."
"Дом" as a building = house, дома́ - a plural of the "дом"
We bought a new house = Мы купили новый дом
I live in this house = В этом доме я живу
"Дом" as a living space or a geographic area (country, town, village...) = home
Home is where my heart is = Дом там, где моё сердце
"дóма = to be at home
Yes, in this context the "дом" (nominative) is wrong, you should use the prepositional case of the word "дом" because the sentence means "mom is — where? — at home" (Мама — где? — дóма).
from Olimo's comment (see above): "When you talk about home, grammar is irregular. (At) home = дома, (go) home = (идти) домой. You have to remember these forms." - and this is exactly right
That would mean "Mom is a house.", so I think you need to put "дома" to get "Mom is home" which means "Mom is at home." Otherwise, there is another Russian sentence "Мама в доме." which I believe means "Mom is in the house."
дома is also the plural for homes and houses in both Nominative case and Accusative case. It is pronounced slightly differently, placing the accent on the a, I think. Yes, the nouns do change forms for different cases.
Now when you look it up in a dictionary, you will see it under "дом". You should check for the declension: https://en.wiktionary.org/wiki/%D0%B4%D0%BE%D0%BC#Russian
Scroll up to Olimo for more information.
The difference between "мать" and "мама" is the same as between "mother" and "mom". The "мать" is more formal and the "мама" is more familiar word for mother.
More about this topic: