This post explains the notation. If you need to see examples, I suggest Sakurai. (My presentation of the notation is much better than his, but I didn't really include any examples of calculations in bra-ket notation).
If you're looking for an explanation of "eigenvectors" of operators with continuous spectrum, I would have to recommend something else. (Some people would say that this is a part of bra-ket notation, but I wouldn't. You'd have the same issues with those "eigenvectors" no matter what notation you're using).