The cases: what are they?

In Russian, words such as nouns and adjectives change their endings according to the role they are playing in the sentence. This is a feature of an inflected language: one which shows the grammar in inflections, or changes to the endings of words. There are 6 cases in Russian, some of which exist in English, although there are very few changes to words to show case in English.

Nominative - this shows the subject of an action, eg. I go, he sleeps, she eats a lot, they play tennis

Accusative - this shows the direct object of an action, eg. he eats an apple, they saw her, we invited them, don't ask me

Genitive - this shows the ownership of a thing, and is equivalent to 's and of in English, eg. her cup, the dog's food, the car belonging to the teacher, the cover of the book

Dative - this can show the indirect object of an action, eg. I gave the book to him (where the book is the accusative direct object); but it is also used after certain prepositions (eg. towards, except for, etc)

Instrumental - this doesn't exist in English, but roughly speaking it tends to be used to express the way in which an action is carried out, or how it is done, but it is also be used after certain prepositions

Locative, also called Prepositional - this doesn't exist in English either, but is generally used after prepositions that express the place of something, eg in the box, on the table. (I prefer to call it Locative, as all the cases can be used with prepositions, so the term Prepositional can be confusing.)