Question

What is the difference between Nom, Akk, and Dativ cases? How can you identify the case? Please give examples!

Answer 1

The cases can be very confusing at times, but once you get the hang of it, you'll identify it more clearly.

The subject does the ACTION.
The subject is the same word as nominative (nom).

The direct object is affected by the verb.
The direct object is the same word as accusative (akk).

The indirect object is the receiver of the direct object.
The indirect object is the same word as dative (dativ).

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EXAMPLE:
I see the book.
I have a book.

GERMAN TRANSLATE:
Ich sehe das Buch.
Ich habe ein Buch.

Ich = I 
sehe = see
das = the
Buch = book

Ich = I
habe = have
ein = a
Buch = book

German is a different form of English in terms of the order of words in a sentence.
In English, "I see the book" - the subject is first, the verb is second, and the object is third.

In German, you can say "I see the book" OR "the book see I." 
You can say it either way.

I see the book = Ich sehe das Buch
Referring to our Yoda impression, "the book see I," is translated into "Das Buch sehe Ich."

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ANOTHER EXAMPLE:

Ich sehe das Fenster.

That new word means window.
Remember, Fenster = window.
Now, refer back to our key words and translate them into that sentence given.

The English translation of the sentence is "I see the window."

The subject in the sentence is "I" since that is the person doing the action.
The object in the sentence is "window" since it is being affected by the action.

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ANOTHER EXAMPLE:

Der Mann siht die Frau.

Der Mann = the man
Die Frau = the women
Sehe and sieht = to see

English translation is "The man sees the woman."
Die and Der means "the" when placed in front of a word. So, technically, the German word for man is mann and the German word for women is frau. 

ALTHOUGH, der and die also mean different things. Der can mean masculine (put in front of man), and die can mean feminine (put in front of woman). One that we did not include was das, which can mean neutral. 
So basically, der, das, and die are all gender forms in German. 

Remember, the subject is nominative and the accusative is the object.
The man would be the subject and the woman would be the object.

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ANOTHER EXAMPLE:

Wir folgen den Zug.

This translates to: We follow the train.

Wir = we
folgen = follow
den = the
Zug = train

NOW, the dativ of this sentence is the indirect object and is the RECEIVER of the direct object. 'Receiver' is a significant word so you can identify it easier. 

So, the dativ in this sentence is 'the train', since they are /following/ the train. What are they doing? They are FOLLOWING the train. Following is also an important word to identify a dativ.

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I'm a bit rubbish on my German, but hopefully this helps you somewhat.