The Min: The Min twists accepted doctrine into something foreign. "Ziftechem Toratchem, v'lo alitchem b'yadchem klum!" - They have falsified their Torah, but gained nothing. Christians would be counted as Minim. A plurality of powers, assigning G-d a corporeal form, etc are all examples of miniut. There aren't very many Min-kofer-bloggers, even though there are some mean kofer-bloggers.
The Kofer - The Kofer denies some important doctrine, such as the existance of G-d. Or the binding nature of the mitzvot. Or that G-d can revive the dead.
The Apikorus - Classically, an Epicurian, but not the ethical-hedonistic values of the Epicure. Rather, the Apikorus denies the immortality, or the existance of, the soul.
The Meshumad - The category that looks at one's actions. Thus, this is one with actual Halachic ramifications. There are several degrees of meshumad, each more severe than the last.
1.One who habitually, due to their desires, violates a comandment
2. One who, out of rebellios spite, violates a commandment
3. One who brazenly, publically, and willfully violates Shabbat
4. One who turns to avodah zarah (Worship of Foreign Gods)
5. The meshumad with regard to the entire Torah. One who drops the whole Judaism shtik. As Rav Lichtenstien quotes the Rambam on this last category:
One who turns to Gentile ways when they are pressed and adheres to them, and says, "What profits it me to adhere to Jews who are lowly and oppressed; it is better for me to adhere to those whose might is superior."As we can see, there is apostasy of thought and apostasy of action. Usually one leads to another.