Case is a grammatical category. It is a kind of inflection - change in the form of a word. The form of nouns, adjectives, pronouns and numerals is changed to express a specific syntactic function. To put it simply, case indicates how one word is related to another one in the sentence. We call this type of inflection declension (deklinace).
Declension also exists in English even though it is not used as much. An example of this is: he - him (I gave him the book.).
In Czech, declension of adjectives and nouns is done by changing the ending. Let's see one example of this: To je moje maminka. (This is my mum). Here, the word maminka is the subject of the sentence and therefore it is in nominative.
On the contrary: Anna vidí maminku. (Anna sees a mum). Here, you can see the ending changed (from -a in maminka to -u in maminku). This means the word's case changed. It is not the subject of the sentence anymore but an object. The subject here is Anna. The mum is seen by Anna and therefore mum is the object of the sentence. The word maminka took the form of accusative (maminku).
Now you are asking, what if we said Maminka vidí Annu., right? As you can tell from the endings, the roles and forms changed. Maminka is now in nominative (subject), Annu is in accusative (object). The sentence would translate as Mum sees Anna.
What cases are there?
There are 7 cases in the Czech language. See the list below. Each case has its own page with explanation of when and how to use it.