Grammatical cases are a common stumbling block for native English speakers learning another language. This might be because cases are sort of invisible in English and they're not taught in school so it's it's hard to see why they would even matter.
However in languages like Czech cases are a really important concept, and if you don't wrap your head around them you'll struggle to make yourself understood. To fully comprehend this importance we need an example - Petr and Pavel, who are not friends.
Accusative - Peter is punching Pavel
In Czech the verb "to punch" is "bít", whose third person conjugation is "bije" so if we want to describe this situation in Czech we might start with something like this ...
Petr bije Pavel
However this isn't quite enough because it's not actually clear who is punching who. You can rearrange this sentence in many ways in Czech, so our sentence can also read Pavel bije Petr and bije Pavel Petr etc. So how do we tell our Czech friends who is the puncher and who is the punchee? Through the magic of cases!
In Czech we indicate the subject (puncher) using the nominative case. In our sentence this is Petr and the nominative case of Petr is simply Petr.
The object (the punchee) of the sentence is indicated using accusative case - which for Pavel is Pavla. Which gives us:
So hopefully you can see why exactly cases are so important, if you don't learn them you're going to confuse a lot of Czech people and have a lot of frustrating conversations. Let's take Petr and Pavel and explore some other cases.
Petr bije Pavla
Genitive - Petr reads Helena's book
Petr čte knihu Heleny
Subject = nominative of Petr = Petr
Verb = third person singular conjugation of čist = čtu
Object = kniha in accusative case = knihu
Possessor = Helena in genitive case = Heleny
Dative - Petr reads to Pavel
If he decides to read the book to Pavel in a curious attempt at reconciliation, we need to use the Dative case:
Petr čte knihu Pavlu
Receiver = Pavel in dative case = Pavlu
Instrumental - Petr reads with Pavel
If this reconciliation is successful and Petr reads the book with Pavel we need to use the Instrumental case:
Petr čte knihu s Pavlem
Instrument = Pavel in instrumental case = Pavlem
Locative - Petr reads about Pavel
Maybe it's weird to describe Pavel as the "instrument", but just go with it because in the next sentence Petr is reading a book about Pavel and in this situation we use the Locative case:
Petr čte knihu o Pavlu
preposition "o" (meaning "about") requires locative case of Pavel = Pavlu
Vocative - Petr says goodbye to Pavel
Finally, to draw this ridiculous situation to a close Petr says goodbye to Pavel where we use the Locative case:
Na shledanou Pavle!
addressing Pavel requires vocative case of Pavel = Pavle!
This is only a quick-n-dirty summary restricted to the seven Czech cases, but it should indicate why each case is important. The Ústav pro jazyk český Akademie věd ČR (Language Institute of the Czech Academy of Science) have a really useful page if you want to see what a word looks like in different cases: http://prirucka.ujc.cas.cz. Just enter your word into the "Slovníková část" section and hit "Hledej"