The nominative case (именительный падеж) is the subject of the sentence. When you see Russian nouns and adjectives listed in dictionaries or on this site, they will be in the nominative case.
With the nominative case alone you can form many simple sentences. For example:
Кто это? Это Иван. = Who is this? This is Ivan.
Ребёнок спит. = The child is sleeping.
Москва - столица России. = Moscow is the capital of Russia.
Сказка рассказана бабушкой. = The fairy-tale is narrated by the grandmother.
Uses of the nominative case
Denoting the subject of an action or state
Мой отец читает книгу. = My father is reading a book.
Introduced by "это"
Это мои дети. = These are my children.
Introduced by "вот"
Вот моя сестра. = Here is my sister.
Used in possessive constructions
У него есть брат. = He has a brother (lit. To him there is a brother).
Меня зовут Aндрей. = My name is Andrei (lit. They call me Andrei).
Used in comparative constructions after "чем"
Я старше, чем мой друг. = I am older than my friend.
Used in generalized constructions after "как"
В таких странах, как Великобритания, книжки-раскраски для взрослых являются бестселлерами. = In countries like the UK, coloring books for adults are bestsellers.
Used in definitions
Лондон - столица Великобритании. = London is the capital of the UK.
Used in apposition to various generic terms
Я читаю роман «Братья Карамазовы». = I am reading the book The Brothers Karamazov.
Nominative case gender word endings
As a general rule, the endings of nouns in the nominative case correspond to the three genders.
Masculine noun endings
Nouns that end in a hard consonant, like завод (factory).
Nouns that end in -й, like музей (museum).
Masculine nouns that end in -ь, like словарь (dictionary).
'Natural' masculines that end in -а/-я, like папа (dad), дедушка (grandfather), дядя (uncle), мужчина (man).
Male names, like Илья (Ilya) an Саша (Sasha).
Diminuitives, like домишко (little house).
Feminine noun endings
Nouns that end in -а/-я, like лампа (lamp), неделя (week), история (story).
Feminine nouns that end in -ь, like ночь (night).
Neuter noun endings
Nouns that end in -о, like лицо (face).
Nouns that end in -е, like море (sea).
Nouns that end in -мя, like имя (first name).
Nominative case plural word endings
There are grammar rules that govern how the noun ending changes from singular to plural depending on the endings of the noun.
One important thing to note, typically nouns that end with a consonant or "-а" require "-ы" to form the plural. However, there are important exceptions. If the word ends with the letters "-г," "-к," "-х," "-ж," "-ч," "-ш," "-щ" then the plural form must end with "-и" instead.