The use of the Genitive Case

(1) The genitive case indicates possession:
дом бра́та – (my) brother’s house
маши́на Та́ни – Tanya’s car
да́ча друзе́й – (our) friends’ dacha/countryside house

(2) The genitive case is also used to indicate relationships between people:
член па́ртии – a member of the party
подру́га до́чери – (our) daughter’s (female) friend
помо́щник президе́нта – the president’s assistant

(3) The genitive is also used in constructions indicating the part of a whole: 
кры́ша до́ма – the roof of the house
коне́ц фи́льма – the end of the movie
кусо́к то́рта – a piece of a cake

(4) In constructions containing two nouns the genitive can indicate the performer of an action:
выступле́ние арти́ста – the artiste’s performance
ле́кция профе́ссора – the professor’s lecture
фотогра́фии друзе́й – the friends’ photos (the ones that they have taken)

(5) The object of an action or process:
убо́рка урожа́я – the gathering in of the harvest
ограбле́ние ба́нка – a bank robbery
фотогра́фии друзе́й – the photos of (our) friends (the ones that depict them)

(6) The second item in a comparison:
Он моло́же бра́та. – He is younger than his brother.
Ната́ша ста́рше Мари́ны. – Natasha is older than Marina.
Соба́ка бо́льше кота́. – A dog is bigger than a cat.

(8) The object of regret (constructions with жаль):
Мне жаль де́нег. – I grudge the money
Мне жаль моего́ дру́га. – I’m sorry for my friend.
Мне жаль тебя́. – I feel sorry for you.

(9) The genitive is used in constructions indicating the quantity of a particular substance:
литр молока́ – a liter of milk
буха́нка хле́ба – a loaf of bread
килогра́мм мя́са – a kilogram of meat

(10) The genitive is used with:
Ма́ло ‘few’, мно́го ‘much, many’, нема́ло ‘not a little’, немно́го ‘not much, how many’, не́сколько ‘a few’, ско́лько ‘how much, many’, сто́лько ‘so much, many’:
мно́го де́нег – a lot of money
ско́лько дере́вьев? – how many trees?
не́сколько студе́нтов – a few students

(11) The genitive is used with negative forms of the verb БЫТЬ (and other verbs with a related meaning) to indicate absence or non-existence: 
Моего́ бра́та нет до́ма. – My brother is not at home.
Книг нет на столе́. – The books are not on the table.
В го́роде нет воды́. – There is no water in the city.

(12) The following verbs are normally used in genitive with with nouns designating abstract and indefinite objects:
боя́ться – to fear, be frightened
добива́ться/доби́ться – to strive for, to attain
держа́ться – to keep to
достига́ть/дости́гнуть/дости́чь – to achieve
жела́ть/пожела́ть – to wish, to desire
избега́ть/избежа́ть – to avoid
лиша́ться/лиши́ться – to be deprived of, to lose
каса́ться/косну́ться – to touch, to concern
ослу́шиваться/ослу́шаться – to disobey
приде́рживаться – to hold to, to keep to
слу́шаться/послу́шаться – to obey
тре́бовать – to demand
хоте́ть – to want
ждать – to wait

(13) The genitive is usually used with the following prepositions: без, для, до, из, и́з-за, и́з-под, кро́ме, на, от, с, со, у, о́коло, вокру́г, недалеко́ от, позади́, напро́тив, посреди́, ми́мо, вдоль. However you should know that the genitive is not the only case used with these prepositions.
чай без са́хара – tea without sugar
о́коло до́ма  – near the house
недалеко́ от запра́вки – not far from the gas station

(14) The genitive case is used for the singular form of cardinal numbers 2, 3, and 4. It is also used for the plural form of cardinal numbers from 5 onwards. (2, 3, 4 – take the genitive singular (any number that ends in 2, 3, or 4 also takes the genitive singular). Anything other than 1, 2, 3, or 4 takes the genitive plural.
2 кота́ – 2 cats
142 кота́ – 142 cats
16 кото́в – 16 cats
55 кото́в – 55 cats