The word is падеж / pl. падежи
Generally there are 6 cases (there's some more, but don't bother with the rest right now):
Именительный: падеж / падежи
Родительный: падежа / падежей
Дательный: падежу / падежам
Винительный: падеж / падежи
Творительный: падежом / падежами
Предложный: о падеже / о падежах

The bad news - judging from the experience of other forum members there's no 'easy' way to know them. Of course, Russian children (whose native language is Russian) do it faster because they intuitively use them already and there is a mnemonic rule, but unfortunately this rule means nothing for non-natives, so the best way to know them is to remember many rules.
Generally a word is changed from one case to another by using the word ending which in turn depends on the gender of the noun or adjective.
Many groups of words, many rules and the most awful part - many exclusions to the rules.
So, what can I help you with - a little bit strange advice, but the best help you can get is to seek help of the non-native Russian speakers. They will really be more help to you than the natives (well, maybe I am wrong about this, but certainly I really do not even know where to begin the explanation).