MATE
First there is the simple straightforward way used as a predicate noun: i.e., "He's my best mate." While in the US we don't use the word this way, I understand it to have the connotation of "friend."
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There is another, more nefarious way this word is used: as a noun at the end of a sentence. As in "Where are you going, mate?" In this case, it appears to be used as a way of softening what might otherwise be a harsh question or comment. Almost like an endearment it can get you out of trouble. For example, an Australian observed of someone who tends to come in later that he keeps "German's hours". I said, "Really? In the US we call those 'Australian hours'!" Everyone just stared at me, silent. So I said, "Uh, I guess I should have said 'In the US we call those Australian hours', mate!" and they all agreed THEN it would have been funny.
"NO WORRIES"
Probably the most overloaded and complex of all the phrases in Australia, learning to decipher this one when you hear it will save you heaps of trouble. And they use it all the time (even more than the word "reckon"). Essentially there appears to be three ways this phrase is used:
Perfunctorily: As when you say in a restaurant, "Would it be possible to have beet root on my burger?" "No worries. Anything else?"
Soothingly: "Oh NO! I meant to clean up that water. Someone might have slipped." "No worries! There's no one here but you right now."
F*ck You, A**hole: "Get that report to me right NOW! Can't I trust you to do anything right?" "No worries. I'll get it."
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No worries.
- Farmer Ted