English language does not have cases, that's why I guess it's hard to understand what they are at all. Basically they have a similar use to that of articles and some prepositions in other languages. Poles get to know them by instinct, I don't even dare try to explain it for we have no idea about grammar rules for that. I find quite interesting the book given in this thread: http://www.italki.com/question/285062 It has a 'cases' section so give it a try. It's a good idea to order your nouns in some tables or whatever you prefer and always make a comment: mówić + D o + I, iść do + A etc.

You need to understand the grammatical functions of each of the cases and how to use them. There's a lot to learn, but don't focus too much on the details, just get an idea of the big picture. For example, I think of the dative as roughly equivalent to the English word "to", as in, I (subject) gave the book (object) to my brother (indirect object). In Polish you would use Nominative for subject, Accusative for object and Dative for indirect object. Yes, some verbs use genitive for their object rather than accusative... they're basically exceptions. You can thinking of potrzebuję as meaning something like "I have a need of..." where "of" reminds you to put the noun into genitive. But again, it's probably easier and faster to just use accusative with every verb and get corrected until you eventually learn instinctively which verbs use which.

The other answers to this question are pretty good, but just want to add a little from my own experience. Firstly, don't try to memorise the tables of noun declinations by heart. I did this, and it still doesn't really help in real situations because you have to conjure up an image of the table in you head and select the appropriate form - which just isn't possible in the timescales of a real conversation. The best way to learn all of the declinations is just to try speaking with a native speaker and choose an ending at random. Or just say it in nominative. Chances are you'll still be understood, and if you're lucky they'll correct you. After being corrected four or five times, your brain automatically starts making the link, and it will be more intuitive and natural than attempting to memorise tables. However, it is important that you understand the grammatical functions of the cases (why we use genitive in certain scenarios) and what each of the cases endings look like - so I'm not saying to stop studying grammar, just not to get bogged down in trying to memorise details. It is possible to figure out the appropriate case to answer a question. The clue is in how the "co? kto?" questions decline, as each of the variants (kogo, czym, czemu) resembles the masculine adjective declination form. So for example the question: "Czym interesujesz się?" could be answered by "polskim filmem", where the form of the adjective polski ends with -im/-ym to match the form of co (czym). This doesn't help you much with muzyką, because it's a feminine noun and therefore its adjectives decline in a completely different way that doesn't match the czym question.

Interesuję się muzyką. You can say: Interesuję się -kim? czym?- muzyką. You should find one word in the sentence wchich responds to any of these questions

There are some rules - on the Internet it looks that they are described a bit. However for a Polish speaker it is just natural, it sounds bad when you use wrong case. But many Polish people do mistakes quite often too regarding cases. For example many people would say for "I'm reading this book" -> "Czytam tą książkę" which is wrong because one should say "Czytam tę książkę", tą->tę. So as Paweł has written, just practise and with the time you'll get some intuition about this.