General explanation: Russian nominative case is equivalent to English one, genitive case is equivalent to English possessive case. Examples:
Этот торт очень вкусный - This cake is very tasty. (Nominative)
Вкус этого торта напоминает мне моё детство - The taste of this cake (this cake's taste) reminds me of my childhood. (Genitive-Possessive).
However, there are some differences:
1. The negation words like "нет", "не был", "не будет" etc. in the meaning "there is/was/were/will be" or "someone has/had/will have" require genitive case.
У меня нет времени (gen.) - I have no time.
2. Some prepositions like "для" - for, "без" - without, "до" - before, "кроме" - except, "от" - from, "около" - next to, "вокруг" - around, "позади", "посреди" - in the middle of, "мимо" - by etc. require genitive case.
Кроме тебя (gen.) все здесь. - Everybody is here except you.
3. Some verbs require genitive case - просить - ask (for), хотеть - want, желать - wish, требовать - require, искать - seek, look (for), ждать - wait (for), достигать - achieve, бояться - be afraid (of).
Ты боишься собак (gen.)? - Are you afraid of dogs?
Most of the Russian nouns with -a ending in nominative are feminine. Exceptions: the words that describe men - папа, дедушка.