Hmm. I will try to answer this as best as I can. There are six cases used in Russian. They are: Nominative, Accusative, Genetive, Prepositional, Dative, and Instrumental.
When would you use them?
(The main uses, I am not writing all the ways the cases are used)
1. Nominative: For the grammatical subject of the sentence
Она студентка. She is a student.
2. Accusative: For a direct object of a transitive verb
Я купила книгу и чай. I bought a book and tea.
“Я” is the grammatical subject and is in the nominative case, “книгу & чай” are the direct objects because they are the object affected by the verb “купила”.
3. Genitive: For possession, to denote nonexistence, relationship between two things, accusative of animate nouns, quantity, after certain numbers, negation after transitive verbs, and after certain prepositions.
Он сын моей сестры. He is the son of my sister
Here we are showing possession. The construction “of” in English is often used with the genitivie case in Russian.
У меня есть кошка. I have a cat.
When you say you, or anyone else, has something, the construction “У” is followed by the genitive case. “Кошка” is nominative because it is the subject.
Нет времени сегодня. There is no time today.
Here we have a case of nonexistence shown by the word “нет”. Время becomes времени as it takes the genitive case.
Есть две ручки. There are two pens.
Numbers in Russian are tricky. Numbers 2-4 take the genitive singular, 5-20 take the genitive plural as well as 5-0 after that. Since the number 2 “два” takes the genitive singular, ручка becomes ручки.
Я вижу моего брата. I see my brother.
Here the direct object after the verb “вижу”is the brother. Normally, we would say it is the accusative case, but since the brother is an animate object, it takes the genitive case instead. мой брат becomes моего брата.
Я спала после работы. I slept after work.
Here we have a genitive preposition, после. Since the preposition takes the genitive case, we must put работа into the genitive.
4. Prepositional: Used after certain prepositions, notably в, на ,о/об.
Я в парке. I am at the park.
Она думает о школе. She thinks about school.
5, Dative: For indirect objects of a verb, impersonal constructions, age.
Анна дала ему новую книгу. Anna gave him a new book.
Anna is the subject, the book is the direct object that goes with the verb “дала”. The indirect object here can be thought of “to who” the book is being given. She gave the book to him so that is why ему is dative.
Мне 5 лет. I am 5 years old.
Here we are expressing age. To say how old you or anyone is, the subject takes the dative case.
Мне надо купить хлеб. I need to buy bread.
Here is an impersonal construction. Expressions such as надо & нужно take the dative case for the subject that needs to do something.
6. Instrumental: to express the means by which an action is performed, with the preposition “c”, predicative instrumental, past of “to be”, time of day.
Она пишет карандашом. She writes with a pencil.
Here we are expressing the means by which the action (writing) took place. With what does she write with? A pencil.
Я смотрю фильм с Анной. I watch the film with Anna.
This is an example of the preposition c. You can think of the construction “with” in English as something that takes the instrumental case in Russian. (There are some exceptions but you don’t see them when you begin to learn Russian)
Я отдыхаю летом. I rest in the summer.
Летом is in the instrumental case as it is a time expression that denotes the time of an action.
Он был врачом. He was a doctor.
Here the past tense of to be “был” in conjunction with an occupation requires the instrumental.
There are many other uses for the cases, but you learn them much later on. There’s also technically a seventh case, but it is rarely used and you don’t learn it.
How do you form them?
Masterrussian.com does a good job of showing you how to do this as it varies depending on gender, number, and spelling. Link: here
How to memorize them?
I learned them by having a declension chart handy. Here is a link to a PDF file. As you start to learn the cases, it helps to keep looking at the charts and relying on them to see what declension to use. Here is an even more simplified chart that is color coded. Make it your best friend.
My first tip would be to make sure you know the gender of nouns in the nominative case. Gender tends to be transparent in Russian, but it can get tricky with the plural of neuter nouns that look like feminine nouns or masculine nouns like папа.
Next, try to remember the common ways that the cases are used. If you see a “в” in the sentence, it probably is the prepositional case; a “с” might mean the instrumental case; нет means the genitive case. I found that this chart that visually shows the cases helps you remember when to use the cases. By having “keywords” to look out for or verbs that you know use certain cases, you create connections between cases and how they are used.
If you look at any declension chart, you can see that many cases share the same endings. This makes it easier by having fewer endings to remember and by reinforcing the same endings in your memory. This post helps with this.
Learn the cases ONE at a time. In a classroom setting, this is simple as the instructor plans the lessons this way. When you self-study, it is easy to try to do all the cases at once, but it will just become confusing. Try the easier cases first such as accusative and prepositional with simple sentences. The more you repeat exercises, the more you will start to “feel” the pattern, making it easier to learn. It takes time so you won’t feel comfortable with cases for a few months, but that is perfectly fine :)
Use an app to help. This app works by presenting you declension exercises. Duolingo does a good job with practice for cases, as well as this Russian grammar app. Link: [Android] [iOS]. Or use a website like this for practice.
I think it helps to read sentences in Russian and learn to identify cases. That gives you a frame from which to work from. If you like Memrise for vocabulary, there is a course that helps you with declensions in Russian to make it a bit more interactive. Really, the way you learn cases is through practice! :D