The common way the apostasy is understood is in terms of the loss of priesthood authority. Priesthood seems the key thing that needed to be restored by Joseph Smith. Much of our conception of restoration is tied to rites and ordinances revealed by Joseph Smith and administered by the priesthood. To my eye the most interesting question relative to the apostasy concerns the Aaronic priesthood. Was it removed from the earth?
The main arguments for it being lost come from John the Baptist restoring it to Joseph Smith in D&C 13:1 where it describes the prayer of this restoration.
Upon you my fellow servants, in the name of Messiah I confer the Priesthood of Aaron, which holds the keys of the ministering of angels, and of the gospel of repentance, and of baptism by immersion for the remission of sins; and this shall never be taken again from the earth, until the sons of Levi do offer again an offering unto the Lord in righteousness.
The word “again” tends to presume it was taken. However this has been seen in different ways. Historically the Aaronic priesthood was just the keys to do various sacrifices and other activities in Israel and was passed down by lineage. Many apologists think that there was a rival priesthood on the Sons of Moses (see D&C 84) roughly corresponding to our notion of Melchizedek priesthood offices. The theory goes that this persisted in Israel in the School of the Prophets at least until the time of Elijah and Elias. It became lost by the time of the exile. While fragments of knowledge of this distinction survived the compilation and development of the Old Testament much of it was redacted and lost.
The question then becomes whether this priesthood restored to Joseph was the Aaronic priesthood as understood by the Sons of Levi or a temporary way of administering in the Aaronic priesthood until “the Sons of Levi” are purified and able to offer up sacrifice again. In particular D&C 13 can be read in that way such that what John gave Joseph wouldn’t be taken away until the Sons of Levi were purified. That is until it wasn’t necessary to have non-Levites administer in these roles. There are numerous scriptures including in the D&C and Book of Mormon saying that the Sons of Levi will be so purified.
If they need this purification though, does that mean they don’t have priesthood until that purification happens? I recognize here that priesthood as a semantic matter is a bit muddled in the LDS thought. Here I’m using it more in the sense of keys of the priesthood rather than in some reified sense of priesthood as independent of practical structural authority. That is I just mean the permission to do something. For the Sons of Levi that means the sacrifices of the law of Moses. Did the apostasy take that permission away?
My guess is that it did not. I think the “purifying” means more an unification of Jewish religion with Mormonism that happens sometime in the future prior to the second coming. I don’t think we can say that if Judaism decided to start up sacrifices again that they would be doing that without permission. Those with a right to the priesthood in Judaism are still reasonably clear. There are even some groups trying to get ready for when that might happen. (This is still controversial within Judaism as I understand it) The Jewish view, as I understand it, is that the purity of the Levites is a ritual purity that has stopped only because the temple was destroyed and those rites are no longer practiced. That is the purification of D&C 13 and other scripture likely is just this ritual purification done by ritual as outlined in the Old Testament. It does not say they don’t have authority to do this just that since the diaspora by Rome, they have not done it.
Given this reading of D&C 13 and Jewish self-conception (Orthodox Jews see sacrifice as being restored by the coming Messiah) can we say there has been an apostasy of the Aaronic Priesthood? I don’t think so. As I said I think we should see the Aaronic Priesthood in our church primarily in terms of Melchezedek priesthood allowing people to function in those roles independent of being a son of Levi. (Much as presumably happened with the Nephites)
 There are some reasons to think the stereotypical way we associate priesthood and apostasy together are misleading. That’s not to deny the importance of priesthood keys just that the apostasy when examined closely seems a bit more complex. BCC had an interesting post on priesthood & apostasy relative to the first vision this week. I don’t want to get too much into the broader questions although there are compelling reasons to think anti-Catholic rhetoric by Protestants shaped the popular Mormon views of apostasy. Not necessarily always in ways that make sense with our own theology. Here though I want to try and keep it narrowed to the Aaronic Priesthood issues.
 According to the documentary hypothesis one of the four main religious movements in Israel was the Priestly one that emphasized Levitical priestly self-conceptions. Given their own views of theology they often contradict other traditions including the ones we’d associate with Lehi from pre-exilic times. If one accepts this view of canon development, at least in general terms, one should be careful with how the priestly source presents themselves in the Old Testament especially against the Sons of Moses or the idea of non-Levites doing sacrifice especially before Sinai. (As quite a few latter-day revelations attest)