Some languages, like German, change the forms of words depending on the function of the words in a sentence. These functions are grouped in categories called cases and German has four of them:
The nominative is used for the subject. This is usually whoever performs the action expressed by the verb. The nominative is also used as the vocative, like when you call someone's name, and it is the form listed in German dictionaries.
The genitive expresses to whom or what something belongs, pertains, from what it originates and so on. It is also used with a few prepositions.
The dative is for the indirect object. This tends to be for whom, to whom, for whose benefit or in some cases to whose detriment the action is performed. The dative is also used with many prepositions.
The accusative is the (direct) object. This is whatever is affected by the action, modified, created, or even destroyed. The accusative is also used with many prepositions.
A simple example sentence may clarify these roles:
The actor's daughter writes the librarian a letter.
Here the subject is ‘the actor's daughter’ - she is doing the actual writing. ‘Daughter’ would be in the nominative, ‘the actor's’ is in the genitive even in English.
The librarian is the addressee. The letter is written to him, so he's the indirect object and would be in the dative in German.
The letter is the thing being written. It is the main thing directly affected by, in this case created by, the action so it's the (direct) object and in German would be in the accusative.
This is necessarily a bit of an oversimplification, but I think it's usually best to start with a bird's-eye view of things. But be aware that in passive constructions the subject is who or what is affected by the action, even though it would still be in the nominative in German. Example:
The letter is being written.
Here ‘the letter’ is the subject, not the object.
As for the actual forms of things, it would go a bit far to list them all here. There are forms for pronouns, articles, adjectives and so on and I'm afraid you'll just have to get used to them. Fortunately, all these things are readily found on the internet.
Because German differentiates between singular, plural and formal you.
.............................. singular ... plural ... formal
nominative ... du ............... ihr .......... Sie
genitive ........... deiner ...... euer ...... Ihrer
dative ............... dir .............. euch ..... Ihnen
accusative ..... dich .......... euch .... Sie
Note that the genitives of the personal pronouns are rare, because for the most common use case of the genitive the possessive pronouns are used instead.
Note also that formal you is capitalised in German.