There are tips and notes for each lesson, only visible in the web app. https://www.duolingo.com/skill/ru/Genitive-Case---1

Tips and notes

In Russian “I have” is expressed by «У меня (есть)» structure. The owner is in the Genitive case.

GENITIVE

"The of-case". It is one of the most universal cases. How do you make the forms? Here is the regular pattern:

TABLE

A zero ending means that the word ends in a consonant or a soft sign (which is just a way to show the final consonant is "soft"). In the Nominative singular, a Russian word can only have the following endings: а, я, о, е, ё ornothing ("zero ending").

GENITIVE OF NEGATION

If you use «нет» to say that there is "no" something or you do not have it, the object is always in Genitive:

У меня́ есть я́блоко → У меня́ нет я́блока

Здесь есть рюкза́к → Здесь нет рюкзака́.

MAJOR USES

"of" (possession): яблоко мамы = mom's apple"of" (amount): чашка чая, много чая = a cup of tea, a lot of tea

A huge number of prepositions requires this case. Yes, «у меня есть», «У неё есть» only use «меня» and «неё» because «у» wants Genitive.

For он, она and оно Genitive doubles as a non-changing possessive "his", "her", "their": его, её, их.

initial «н» is used for him/her/them with the majority of prepositions (doesn't affect possessives)

INDECLINABLE NOUNS

A little side note: some nouns of foreign origin are indeclinable. It means that all their forms are the same. Foreign nouns that end in о/е become like that (кофе, метро, радио, резюме), as well as all nouns that do not fit into Russian declension patterns (see above).

This includes female names that end in anything other than А or Я. A few -ь-ending names are an exception (Любовь and Biblical names like Юдифь).

So, all of the following names are automatically indeclinable: Маргарет, Мэри, Элли, Дженни, Рэйчел, Натали, Энн, Ким, Тесс, Жасмин.

I AM AWAY

Russian also uses the Genitive to state that someone is "away", "not there": Мамы сейчас нет. In English such use would correspond to "There is no mom at the moment", or even "There is no me now". We are not hard on that particular construction in the course, but it is important to know it all the same.

Added bonus: when a verb directly acts on a noun, the noun is called a direct object and is in Accusative. In Russian, only -а/-я feminine nouns have a unique form for it. Others just reuse Genitive or don't change the word at all (Nominative)

NOTHING

Russian uses.... let's call it "consistent" negation. It means that in negative sentences you are required to use "nothing" instead of "anything", "nowhere" instead of "somewhere" and so on. Let's meet the first of these pronouns:

У меня ничего нет. = I don't have anything.Она ничего не ест. = She doesn't eat anything.

You'll also notice that, unlike standard English, Russian has no rule against using double negatives.

Yep, the app seems to have only the quizzes, and no educational material as seen on the desktop site.

The "nonexistent" object preceded by «нет» is always in the Genitive case.

Neuter nouns (those ending in -о/-е) and consonant ending masculine nouns use А (or Я) as their Genitive ending. In the singular, of course.

A few examples:

• NOM: мальчик, отец, учитель, велосипед, компьютер, молоко, яблоко, море -GEN: мальчика, отца, учителя, велосипеда, компьютера, молока, яблока, моря

Elsewhere someone posted this answer to the same question:

«Не есть» is not used for existence and possession. "Нет" is used instead.

So I think that means you use "есть" when someone HAS something, or that thing exists, and "нет" when they DO NOT HAVE something, or it does NOT exist.

Also see response by Theron126 below in this thread.

У is a preposition that "wants" the Genitive. So we use it. Other Genitive prepositions include без, возле, около, от, из, с (in the meaning of point of origin), до, после. There are many others.

Try not to think too much about it. When a preposition or a verb requires a certain case, coming up with a reasoning why this case has such use might be hard for some of them—and would probably require you to follow many centuries of the history of the language.

So-called 7-letter rule (which native speakers, by the way, do not know) places restrictions of which vowels can combine with certain consonants. It does not tell you the endings you should use.

Яблоко and девочка use different ending patterns. Moreover, яблоко has an unusual plural (neuter nouns ending in о or e normally make plurals ending in а or я)

"the" can translate as "это" or it can be left untranslated.

"The boy is not an apple..." You need "у мальчика". It's incorrect to use "не есть" in this sort of sentence, you should instead say "нет". Therefore "у мальчика нет яблока".

не negates a verb and is written before a verb, so it can't come after есть.