In the Torah, Aaron is the first high priest of the Levitical order of priests. He had four sons: Nadab, Abihu, Eleazar, and Ithamar. The first two died in Leviticus 10, leaving Eleazar as the eldest inheritor of the high priestly line. We see in Joshua that Eleazar's son Phinehas had become the high priest by the time of the conquering of the land. Then we lose any record of what was going on with tabernacle worship until we get to Samuel, where there seems to be a fixed temple structure built up around the tabernacle implements of worship from the end of Exodus. Perhaps the most surprising feature of the priestly situation at the beginning of the book of Samuel is that the high priest Eli was not descended from Eleazar but his younger brother Ithamar. Where are the descendants of Eleazar, then? What happened after Phinehas?
This surprising fact in the Samuel history has led a number of scholars to propose a skeptical reconstruction of what really happened. On this view, the high priestly family has always been descended from Ithamar, and Eli's family in Samuel was the original high priestly family. With David we get the insertion of a priest named Zadok alongside the final remaining Elide priest Abiathar/Ahimilech. I Chronicles 24 tells us that Zadok is the head of the the Eleazar clan of priests, which Ahimelech (perhaps the same man called Abiathar in Samuel, perhaps his son) was head of the Ithamar clan of priests. The revisionist theory takes Zadok to be a complete outsider from the conquered Jebusite city of Jerusalem. David allowed him to continue his priestly duties, casting him as a priest under the order of Melchizedek, the original priest-king of Salem (which became Jerusalem) from Genesis 14. This allowed David to assert his legitimacy to be king in the Jebusite city, and then Chronicles and the other places that list him as a descendant of Eleazar are just reworking the tradition to make him fit the Israelite origin story, casting Zadok as a son of an older brother of the ancestor of the Elide priests. Thus no sign was left of the Jebusite origin of Zadok.
My first thought upon hearing this theory is that it seems entirely unmotivated by the evidence. After all, the only texts we have indicate Zadok as Levitical priest in the line of Aaron, Eleazar, and Phinehas. There is a difficult issue about what's going on in the Elide line of the Ithamarites, i.e. whether Ahimelech and Abiathar are the same person or father and son (or whether there's some other explanation why the texts say what they do in a way that seems to conflict), but that doesn't affect the issue of Zadok. The argument for this reimagining of the history of Israel's priestly line is pretty much that you can read the evidence this way if you really, really want to as long as you're willing to deny most of what the text actually says on the issue. That's a pretty flimsy argument, but much 20th century biblical studies is like that. We really have no further evidence besides the text on this issue, so I've never understood the motivation for this view apart from the need to come up with something new to maintain a job as a scholar.
I will admit that the evidence is consistent with this as long as you're willing to be highly suspicious of our best sources on the matter without any clear reason to suspect them of unreliability on this particular issue. The biblical authors took the Torah accounts with utmost seriousness as to their exact details. You have to propose that any such care was politically constructed by those with a deceitful agenda to legitimate an illegitimate Zadokite line if you want to make a view like this work. You could then choose to see the genealogies of a carefully geneaological society as complete fabrications. You would ultimately be telling a consistent historical account of how things went to shape the texts as they are now, but it seems unlikely to my mind that such a far-ranging reimagining of the events is necessary to explain the text as it stands. A much more likely account is available, one that's fully consistent with accepting the events as related in the biblical texts and thus much simpler in terms of not postulating unnecessary things that we have no further evidence for. I haven't seen anyone mention this explanation in any of my reading on the issue, though I'm willing to be corrected on that score if anyone knows of someone mentioning it as a possibility.
After the time of Joshua and Phinehas, the biblical texts present Israel in turmoil. The unity of the people in their arrival in the land rapidly deteriorates. The book of Judges presents a situation in which the priests are not teaching the Torah, and the people are not following it. It may well be that those who followed Phinehas had in fact abandoned their Torah-mandated function of teaching the people to follow and observe the Torah. They may even have abandoned the priestly role entirely. The book indicates a priest-for-hire role that some Levites were occupying, and perhaps this commercial venture was more attractive to the high priestly family. In any case, it's highly likely in such a situation that those who depended on tithes from people who were surely not giving them would seek sustenance elsewhere. Levites did have some use of land around the Levitical cities, and they may well have simply abandoned their task for financial reasons. The ark still existed, and the need for defense from raiding enemies or conquering peoples would in some cases have involved the carrying of the ark (as it seems to have involved in the early chapters of Samuel), and so the few Levitical priests who were able to sustain themselves could easily have continued on in the absence of the biological inheritors of the Aaronite high priestly role, who may have left for more sustainable shores. This would then explain the missing Eleazar line in the time of Samuel.
[Update: Another possibility that occurred to me after I published this post seems almost as likely to me. Perhaps what happened was not the loss of a centralized high priestly family with another family replacing them. Maybe it was just a decentralization, with different priestly families handling priestly responsibilities in different parts of a highly decentralized Israel (which the book of Judges testifies to). The Elide family of the clan of Ithamar might have been a localized priestly group calling Eli their high priest. This would fit well with how the Israelites had decentralized in the period of the judges. Eli is never to my knowledge called High Priest, just "the priest", but that is the same term used of Zadok in David's reign. What makes me lean against this hypothesis is that Eli and his sons did have the ark of the covenant in I Samuel 4, so it's likely that they were the leadership of what priests continued to serve at the temple structure that succeeded the tabernacle. There might be some other explanation for why a family of the less significant priestly line would have the ark of the covenant when the more significant line was functioning as priests elsewhere, but I'm not sure what that explanation would be.]
David's portrait in both Samuel and Chronicles (never mind the Psalms) is that he had a deep-seated commitment to the word of God in whatever form it took. If you take the biblical materials at face value, there was a Torah already when David became king, and a king with the character David is presented with would have had serious concerns for doing things properly, especially after the failure in moving the ark to Jerusalem in II Samuel 6. So would it be surprising for a king of such character to examine the Torah on the high priestly line and seek out the proper high priestly descendant of Phinehas to occupy the role of Aaron? Finding the legitimate high priest is exactly what we would expect of a king with the features the Samuel and Chronicles accounts describe David as having. But the text doesn't emphasize this, which is a good sign that the author didn't make it up for that purpose, or it would have said something about his origins to underscore the point. I think, then, that we should take the text at face value. The one sign we would expect of doctoring the text according to standard analyses that seek to detect ideology isn't really present.
If this is right, then a possibility that seems wholly unsurprising given the biblical texts at face value (the falling away or departure of the Eleazar line and the calling back of Zadok by David to meet the Torah regulations) would explain the one strange element that appears in those texts as taken at face value (the Eli line from Ithamar as high priests in Samuel's time and the appearance of Zadok out of nowhere with no explanation in the text). Why would someone need to appeal to Zadok as an outsider to Israel altogether as if his place in the line of Eleazar was purely fabricated? I don't see the motivation. The traditional view that the documents record is the only view the documents provide any evidence for, and the one objection has a fairly easy solution. I therefore don't see how anyone could think there's enough reason to believe this alternative account, never mind to base a whole reconstruction of Israelite history on it.