The nouns in a sentence each take a different role. For example, in the sentence:
Ich habe ein geiles Motorrad gekauft.
‘Ich’ is the subject (the noun doing the action), ‘ein geiles Motorrad’ is the object (i.e. the noun which is having the action done to it). In German, the grammar surrounding a noun (i.e. the endings on the articles, the adjectives, and so on) changes depending on the role it plays in the sentence, and we therefore say that the noun and the grammar related to it is within a certain ‘case’. It’s a bit like how in a pantomime the pirate king is the ‘baddie’, which means that the audience have to hiss at him and he has to wear an evil-looking mustache, because his character and the actions and costumes pertaining to him are all related to the ‘baddie’ role.
In a simple sentence, the subject is always in the nominative case, and the object is always in the accusative case. You can also have objects which are in the dative and genitive cases. All these separate cases are discussed in the other Glossarama articles hyperlinked here.
Nominative case
Dative case
Genitive case