I'm having some trouble understanding bra-ket notation?

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I have been studying quantum physics for some time, yet I still cannot seem to understand the principles behind bra-ket notation (especially how spin states are described and the probabilities/eigenvalues). It would be great if someone could give me a basic explanation and/or maybe recommend some helpful resources.
 
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This should be explained in any introductory level textbook on quantum mechanics, but I assume you have read one. If that is the case, it would make it easier for us to help you if you specify what exactly in the notation you are having problems with or find unclear. If not, i suggest starting with a textbook.
 
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