The expression hysteron proteron consists of two Greek words, hysteron meaning 'later' and proteron, 'before, earlier'. So hysteron proteron is an inversion: it involves putting what should come later before what should precede it. The expression has two main uses, in the second of which it is always a term of disapproval.
In the study of literature and similar areas, hysteron proteron involves words being put in an unexpected order for effect. (This is what makes it a deliberate figure of speech.) For example, many writers vary the way they report conversation by using the construction "says he". This is an example of hysteron proteron: the usual order in English is Subject and then Verb, but here we have Verb + Subject. The effect is sometimes to give a slight feeling of doubt. (Compare this type of (deliberate) hysteron proteron with that involved in someone's accidentally saying things 'the wrong way round', as when the harassed parent says, "Oh go to sleep and go to bed." The child might feel that this was an example of hysteron proteron, wanting, naturally, to go to bed before it goes to sleep.)
In the study of logic, hysteron proteron is the name of a fallacy in which the proposition which is to be proved (and which should therefore be the conclusion of the argument) is assumed as one of its premises.