"Mom does not have a brother."
Translation:У мамы нет брата.
The latter terms are in the genitive case, used in certain grammatical constructions. A lesson on noun declension is needed to fully understand as it's not a concept that appears in English.
Yes, the word following нет must conjugate to the genitive case. «У меня есть брат» ("I have a brother") vs. «У меня нет брата» (I do not have a brother). Do you understand? ☺
From what I understand genetive works like this: "У [Genetive] есть [Nominative]" The "owner" is the only word that changes. So "Брат" is still in the nominative case, but it changes because of the negative "Нет"?
These are forms of the same word. «Мамы» is the Genitive one.
У is the prepositions that always wants the Genitive (regardless of the exact meaning), so you use it here. Using the у+Gen (+ «есть») phrase is the most common and ubiquitous structure to express possession in Russian, so we have it early on.
I'm not following where we use "не" and where we use "нет". Is it gender, or something else?
«Не» generally corresponds to "not" and we use it to negate a member of the sentence. «Нет» usually means "no" and we use it to start a negative answer or to express a lack of something. Нет can work as a short negative answer (Ты спишь? — Нет.). Alternatively, if you are answering a question that has a verb, you can use не + the form of this verb that you would use in a dull fledged sentence (Ты спишь? — Не сплю). Of course, we do not have dialogues in our course.
We also use нет when omitting the part of the sentence that gets negated (but would it be repeated if it were left). For example, "Я переводчик, а моя жена нет" (I am a translator and my wife isn't) or "Они спят, а мы — нет" (They are sleeping and we aren't).