Theological answer
Naskh, an Arabic language word usually translated as "abrogation" and alternately appearing as the phrase al-nāsikh wal-mansūkh ("the abrogating and abrogated [verses]"), is a technical term for a major genre of Islamic legal exegesis directed at the problem of seemingly contradictory material within or between the twin basises of Islamic holy law: the Qur'ān and the Prophetic Sunna. In its application, naskh typically involves the replacement (ibdāl) of an earlier verse/tradition (and thus its embodied ruling) with a chronologically successive one. The complete suppression (ibtāl) of a regulation so that not even its wording remains is recognized as well, though only in the case of the Qur'ān.
Quite simply, when two parts contradict each other, which ever one was said by Muhammed last is considered offical. Then the fun starts, because the Koran isn't in chronological order there are verses where you can arrange things so that which ever verse you want is last.
Muslim scholars also use the standard Biblical methods. Hair splitting arguments over trivial differences between verses, careful choice of the amount of context given for verses to obscure it in some cases and detail it out in others. Arguments over which verses are allusions and other literary constructs and which should be read literally. Claims that the scope of some verses are very narrow and apply only to specific people or places or periods in history while other verses have a broad scope and apply always. There are fewer arguments that a word in the Koran meant something different then it does now, but that does come up from time to time.
In fact the only one you don't see are aguments about how something should be translated, because all serious religious discussion of the Koran is done in Arabic. But there is a closely related problem in the recitation of the Koran though. There are several different ways of pronouncing the verses of the Koran because the original arabic doesn't contain vowel information. In must cases these differences are trivial differences, but in some they disagree about the tense of a sentence or other significant feature.
The Koran itself isn't subject to the arguments about which books should be included and which shouldn't, but the Hadith are a critical part of the faith and are subject to even more arguments in that regards then the Bible.
In practice, it works rather like the Bible. People come to a conclusion*, and then they think up verses that support their posistion.
Jay
* Of course the religion also influences what conclusions they come to. People know what their religion considers acceptable and what matters are considered dogma.