1. A 'mate' is just another word for a friend, used mainly in British and Australian English. It can combine with other words to mean various different things. Here are the commonest combinations:
A flatmate = someone who lives in the same flat as you.
A workmate = someone who works in the same place as you
A classmate = someone who studied in the same class as you
2.
Your aunt is your father or mother's sister.
Your great-aunt is your grandfather's or grandmother's sister.
In other words, they are different generations. An aunt is of your parents' generation, whereas your great-aunt is of your grandparents' generation.
An aunt doesn't have to be a blood relative - you also call your uncle's wife your 'aunt'.
3.
Sister-in-law (meaning 1) = your brother's wife
Sister-in-law (meaning 2) = your wife's sister/your husband's sister
Half-sister = This is someone with whom you share one parent.
For example, let's say that your parents get divorced. Your father marries again and has a child with his new wife. That child is your half-sister or half-brother, meaning that you have the same father but a different mothers.
Your father's new wife is your 'stepmother', and if she brings with her a daughter from her first marriage, that daughter is your 'step-sister'.
I hope that all makes sense.