Traditionally there are three main classes of miracle:
1. Supernatural miracles are miracles in the strictest sense. They are worked in subjects in which similar effects never occur in the ordinary course of nature. Thus, it never happens in the ordinary course of nature that after being clearly dead and untouched in a tomb for a day a man comes alive again; only extraordinary intervention could make it possible. (These are also called 'miracles above or beyond nature'.)
2. Preternatural miracles are worked in subjects in which (through the forces of nature, either left to themselves or artificially applied) similar effects do occur in the ordinary course of nature. What distinguish them as miracles are the precise circumstances, which indicate that this is not a case in which we are dealing with the forces of nature left to themselves or artificially applied. Common examples given for this category are the plagues of Egypt: it's possible in the ordinary course of nature for there to be a massive infestation of frogs (second plague) in the ordinary course of nature; it is even possible in the ordinary course of nature for this to occur after a disturbance of the waters (first plague) and to lead after the death of the frogs to an infestation of insects (third and fourth plagues). What indicates the miraculous nature of the plagues is (1) the piling up of the ten plagues on top of each other; (2) the interaction between Moses and Pharaoh; (3) the protection of Goshen. In other words, it's not the facts themselves that are miraculous; it's that they occur in conjunction with a special set of circumstances. (Preternatual miracles are also called 'miracles outside of nature'.)
3. Counternatural miracles are worked in subjects that by nature tend to the contrary effect in the ordinary course of nature. Thus, ordinarily if you throw human beings into a massive fire, they die; if, as with the three youths in Daniel, they are just fine, that's a miracle. (Counternatural miracles are also sometimes called 'miracles against or contrary to nature'.)
In a number of ways the second class is the most interesting. It shows, for instance, that there can be no easy a posteriori argument against the possibility of miracles; and the standard arguments against the possibility of miracles don't refute the possibility of preternatural miracles, even if the arguments are held to be fully sound (Butler has a good argument showing why in the second part of the Analogy). The only way to refute the possibility of preternatural miracles is to prove that there is no cause capable of performing them. At the same time, preternatural miracles would be harder to prove, qua miraculous, than a counternatural or supernatural miracle would be, because the evidence would always be circumstantial. (The fact that they are hard to prove qua miraculous is why the Catholic Church, for instance, only rarely considers them when it comes to canonization procedures -- thus a remission of cancer is not generally treated as a miracle for canonization purposes, even though the Church doesn't rule out the possibility that some remissions that are not officially recognized as miracles may be miraculous.)
It occurred to me after I wrote the above that I should say something about what is not included as a miracle. An occurrence can only be a miracle if it involves signs of divine agency. In the case of supernatural and counternatural miracles, the occurrence itself is the sign; in the case of preternatural miracles, the whole set of circumstances is the sign. But not all divine actions would be miraculous. A good example is impetration. If I pray for x to happen and x happens, where x is some ordinary event (like getting over a cold), there is nothing miraculous about it whatsoever. It does not follow from this, however, that God did not answer my prayer; all that follows is that whatever was done was done according to ordinary providence, not miraculously. Only if x is accompanied by signs indicating special divine agency does it count as a preternatural miracle.
