Formally Russian nouns, adjectives, numerals, pronouns and participles in different cases differ by its endings. For example:
nom. sing. стол (table) -- null ending
gen. стола -- ending is "а"
dat. столу
acc. стол
inst. столом
prep. столе
In sentence different cases perform different functions. For example:
Она дала студенту книгу. She gave the student a book.
Pronoun "she (она)" is the subject, therefore it is in nominative. Noun "student (студент)" is the indirect object, in Russian it is dative (студенту). Noun "book (книга)" is the direct object -- in Russian it is accusative (книгу).