A sentence has to have at least one noun and at least one verb. The required noun is the subject[1] of the sentence, or the thing doing the action of the verb. So "the man writes" is a complete (albeit short) sentence with “the man” being the subject.

If we now ask what he is writing we can introduce another noun into the sentence and say "the man writes a book". The book is now object that the verb action is done to.
These two ways of using nouns (or "cases") have the names Nominative and Accusative where the Nominative case is the way we handle "subject" nouns and the Accusative case is the way we handle "object" nouns.
Now let’s complicate things a little further as there's a lot more than just two cases of nouns. In fact there's a whole "case" of worms! Very briefly we've also got:

• The dative case indicates the indirect object of a verb: The man gave his daughter a book.
• The ablative case indicates the object of most common prepositions: The boy went with his father to see the doctor.
• The genitive case, which corresponds to English's possessive case, indicates the possessor of another noun: A country's citizens must defend its honor.
• The vocative case indicates an addressee: John, are you O.K.?
• The locative case indicates a location: I live in China.
• The instrumental case indicates an object used in performing an action: He shot it with the gun.

The way we handle these different uses of a nouns in English doesn't tend to alter the noun or it's article (the or a/an) as we use word order and prepositions (like to, in, of with etc) to distinguish what we mean. However in Greek this isn't the same. This is one area where you must remember that translations cannot be word for word equivalents and you must understand that different languages have different ways of saying the same thing.
So, lets deal with the Genitive case which, in English, is referred to as the possessive case[2]. As it's name suggests this deals with ownership of another thing. Now there's lots of sub-divisions of the Genitive case but it's best not to go too deeply into that now[3]. Let's just stick to some obvious examples:

• THIS is Dave's webpage.
• The MEN of Sparta fought the Persian army.
• YOU are George's wife.

The Subject is in red (Capitals), the Object is in green (italic) and the Possessive noun in blue (underlined). In these examples we see two different ways in English that the Genitive is used, either by adding an apostrophe and an "S", or by using the preposition "of". In Greek the Genitive is recognised by a change to the article and to the ending of the noun[4]. This also happens in the Accusative case too although in many cases is not as obvious an effect[5].
The following table shows all the changes the happen for both the Accusative and Genitive cases in Greek:

Gender	Singular						Plural
	Nomenative		Accusative		Genative		Nomenative		Accusative		Genative
	Article	Noun Ending	Article	Noun Ending	Article	Noun Ending	Article	Noun Ending	Article	Noun Ending	Article	Noun Ending
Masc	Ο ΕΝΑΣ	-ΟΣ	ΤΟ(Ν) ΕΝΑ(Ν)	-Ο	ΤΟΥ ENΟΣ	-ΟΥ	ΟΙ	-ΟΙ	ΤΟΥΣ	-ΟΥΣ	ΤΩΝ	-ΩΝ
		-ΑΣ		-Α		-Α		-ΑΔΕΣ		-ΑΔΕΣ		-ΑΔΩΝ
		-ΗΣ		-Η		-Η		-ΕΣ		-ΕΣ		-ΕΔΩΝ
		-ΕΣ		-Ε		-Ε		-ΕΔΕΣ		-ΕΔΕΣ		-ΕΔΩΝ
Fem	Η ΜΙΑ	-Α	ΤΗ(Ν) ΜΙΑ	-Α	ΤΗΣ ΜΙΑΣ	-ΑΣ	ΟΙ	-ΕΣ	ΤΙΣ	-ΕΣ	ΤΩΝ	-ΩΝ
		-Η		-Η		-ΗΣ		-ΕΣ		-ΕΣ		-ΩΝ
		-ΟΣ		-Ο		-ΟΥ		-ΟΙ		-ΟΙ		-ΩΝ
Neut	ΤΟ ΕΝΑ	-Ο	ΤΟ ΕΝΑ	-Ο	ΤΟΥ ΕΝΟΣ	-ΟΥ	ΤΑ	-Α	ΤΑ	-Α	ΤΩΝ	-ΩΝ
		-Ι		-Ι		-ΙΟΥ		-ΙΑ		-ΙΑ		-ΙΩΝ
		-ΜΑ		-ΜΑ		-ΜΑΤΩΣ		-ΜΑΤΑ		-ΜΑΤΑ		-ΜΑΤΩΝ
		-ΟΣ		-ΟΣ		-ΤΟΣ		-ΩΤΑ		-ΩΤΑ		-ΩΤΩΝ

Gender	Singular						Plural
	Nomenative		Accusative		Genative		Nomenative		Accusative		Genative
	Article	Noun Ending	Article	Noun Ending	Article	Noun Ending	Article	Noun Ending	Article	Noun Ending	Article	Noun Ending
Masc	ο ενας	-ος	το(ν) ενα(ν)	-ο	του ενος	-ου	οι	-οι	τους	-ους	των	-ων
		-ας		-α		-α		-αδες		-αδες		-αδων
		-ης		-η		-η		-ες		-ες		-εδων
		-ες		-ε		-ε		-εδες		-εδες		-εδων
Fem	η μια	-α	τη(ν) μια	-α	της μιας	-ας	οι	-ες	τις	-ες	των	-ων
		-η		-η		-ης		-ες		-ες		-ων
		-ος		-ο		-ου		-οι		-οι		-ων
Neut	το ενα	-ο	το ενα	-ο	του ενος	-ου	τα	-α	τα	-α	των	-ων
		-ι		-ι		-ιου		-ια		-ια		-ιων
		-μα		-μα		-ματος		-ματα		-ματα		-ματων
		-ος		-ος		-τος		-ωτα		-ωτα		-ωτων

Let’s look at a few examples:

1) I drink in Niko’s bar

The vocabulary we need is “I drink” πίνω (ΠΙΝΩ), "in" σε (ΣΕ), "the bar" το μπαρ (ΤΟ ΜΠΑΡ) and the owner’s name, Νικος (ΝΙΚΟΣ). The subject “I” is implied in the verb. The object of the sentence is “the bar” which is neuter so the article and ending doesn’t have any changes in the accusative (although “σε (ΣΕ)” and “το (ΤΟ)” merge to form “στο (ΣΤΟ)”), and the bar belongs to “Νικος (ΝΙΚΟΣ)” which is obviously masculine so the article “ο (Ο)” changes to “του (ΤΟΥ)” and the noun ending “–ος (-ΟΣ)” changes to “-ου (-ΟΥ)”. So the complete sentence becomes:

πίνω στό μπάρ του Νίκου (ΠΙΝΩ ΣΤΟ ΜΠΑΡ ΤΟΥ ΝΙΚΟΥ)

2) The farmer’s dog bites the man
Our vocabulary is:

• The Farmer = ο γεωργός (Ο ΥΕΩΡΓΟΣ)
• To Bite = δαγκώμα (ΔΑΓΚΩΜΑ)
• The Dog = ο σκύλος (Ο ΣΚΥΛΟΣ)
• The Man = ο άνδρας (Ο ΑΝΔΡΑΣ)

Note that all the nouns are masculine so there are going to be lots of article and noun ending changes. The dog is the subject as he’s doing the biting, so the case is nominative and so we can stick to the normal article and ending. As it’s the dog biting, the verb changes to the third person singular ending (he bites) which is δαγκώνει (ΔΑΓΚΩΝΕΙ). The object is the man as he is the one wounded, so the article “ο (Ο)” becomes “τον (ΤΟΝ)” and the ending “-ας (-ΑΣ)” changes to “-α (-Α)”. Finally the possessive noun is the farmer as he owns the dog, so the article “ο (O)” changes to “του (ΤΟΥ)” and the ending “-ος (-ΟΣ)” becomes “-ου (-ΟΥ)”. The Greek sentence is then:

ο σκύλος του γεωργόυ δαγκώνει τον ανδρα (Ο ΣΚΙΛΟΣ ΤΟΥ ΓΕΩΡΓΟΥ ΔΑΓΚΩΝΕΙ ΤΟΝ ΑΝΔΡΑ)

 

3) The women’s garments have many colours

• The woman = η γυναίκα (Η ΓΥΝΑΙΚΑ)
• The garment = το ρούχο (ΤΟ ΡΟΥΧΟ)
• To have = έχω (ΕΧΩ)
• Many = πολύς (ΠΟΛΥΣ)
• The Colour = το χρώμα (ΤΟ ΧΡΩΜΑ)

This time we are dealing with plurals so follow the chart to see how the articles and endings change to make:

τα ρούχα των γυναικων έχουν πολά χρώματα (ΤΑ ΡΟΥΧΑ ΤΩΝ ΓΥΝΑΙΚΩΝ ΕΧΟΥΝ ΠΟΛΑ ΧΡΩΜΑΤΑ)

Note also that “to have” has to change to 3rd person plural (They Have) and the adjective ending for “many” also has to match the number and gender of “colours”.

Notes:-
[1] Now here is a big difference between English and Greek. In English the Subject is the only noun that is required to form a sentence, even if it is in the form of pronoun. This is normally the first cardinal rule of English grammar. However remember that in Greek, the verb ending will imply the pronoun and so the subject is often missing from (or rather implied in) Greek sentences. For example “He writes” is a very short sentence in English with the subject being “he”. In Greek this sentence is just “γράφει (ΓΡΑΦΕΙ)”, a single word!
[2] There is argument as to whether English actually has a possessive case or not depending on whether the apostrophe & s ending to a noun represents a real noun ending change or simply an abbreviation, for example “Tom’s car” being an abbreviation of “Tom, his car”
[3] There are distinctions between such things as “Tom’s Car” which indicates true possession and phrases like “Men of England” which refers more to the origin of the men but in a way that stresses the identity of being English, rather than the non genitive phrase “Men from England”
[4] It may make more sense if you consider the other theory on why the ’s ending is used in English is that in old-English the genitive was formed by adding es to the end of many nouns. So instead look at “ic scolde æghwær godes lof upp aræran” which is from a proclamation of King Cnut which translates as “I should ayewhere(everywhere) God's love(praise) uprear(promote)” and notice the “es” ending on the word for God.
[5] The Accusative has the most impact in Masculine nouns in Greek where both article and noun ending change. For feminine words generally it is only the definite article that changes and Neuter nouns do not change at all between Nominative and Accusative.