Dative case - celownik

The dative case is used for the indirect object of a sentence.  The indirect object is whom / what the action is being done for.  For example, in the sentence 

Marek is giving the beer to Piotr.

Marek is the subject (so nominative case).

The beer is the direct object (so accusative case).

Piotr is the indirect object (so dative case)

Marek daje piwo Piotrowi.

Another example:

I'm buying the beer for you.

Kopuję ci piwo - ci is the dative form of ty (you) in informal Polish.  Tobie is also the dative form.

The dative case used after verbs that require an indirect object, as you might expect.  Examples:

Dziękować - to thank somebody. 

Mówić - to say to / tell somebody.  I told you...  Mówiłem Ci... (NB if you are female, you would say Mówiłam Ci...)

Odpowiadać - to reply to somebody.

Płacić - to pay somebody.

Podobać się - to appeal to somebody.

Przypominać - to remind somebody of something or to resemble somebody.  Przypominasz mi.

Życzyć - to wish somebody something.  

Impersonal expressions (with no subject)

Gorąco mi / Zimno mi - I feel hot /cold (it's hot/cold to me)

Łatwo ci mówić - Easy for you to say (seems to be used as a retort in the same way as we would in English)

Smutno mi - I feel sad (it's sad to me)

Jest mi słabo - I feel faint

Głupio mi - I feel stupid

Nudzi mi się - I'm bored

After the prepositions:

Dzienki - thanks to.  

Dzięki Bogu!  Thanks to God!  

Dzięki Kingi mówię trochę po polsku.  Thanks to Kinga, I speak a little Polish.

Wbrew - contrary to / despite / against.