The dative in German is used to describe an indirect object, for example:

I gave the horse an apple
^ Nom   ^Dat     ^Acc

Nom: Nominative
Dat: Dative
Acc: Accusative

and after the prepositions:

vor
hinter
über
unter
zwischen
neben
in
an
auf

It is used when no motion towards is conveyed, whereas the accusative implies motion, eg:

Es sieht besser aus unter dem Fenster, nicht?

It goes better under the window, no? ~=~ Doesn't it look better under the window?

It is always used after these prepositions:

aus
bei
mit
nach
seit
von
zu
gegenüber
außen

Also, certain prepositions can be contracted, some common ones are listed below:
an dem  -> am  
in dem  -> im
zu dem  -> zum
zu der  -> zur

Gender  Nom Dat
Masc    Der Dem
Fem     Die Der
Neut    Das Dem
Plur    Die Den

Beware, the next section will haunt your dreams and plague you in hours of daylight whenever the case system is mentioned.

An easy way to remember the dative is by using the mnemonic:

Monkeys Raisins Monkeys Nuts

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