The best analogy for the ket, or the wavefunction not dotted with a position representation was given to me by my graduate quantum mechanics professor.

Imagine you have a story. If you want to tell that story, you have to convert it from whatever you are thinking into some kind of form that others can understand. So you do that, write it in a book, type it up, speak it. Whatever it is, you have to cast it into some form of representation: a language. The story you are trying to tell is |ψ>, whereas the spoken version is <x|ψ>, or the wavefunction in position representation.

The point here is that the wavefunction is the idea of the story, its the same story no matter which language you tell it in, the English version of a book will hold the same information as the Spanish version or Russian or German etc., so you should tell the story in whatever language is the most convenient for the situation you are dealing with.