How are Quranic and hadith texts that appear to contradict each other reconciled?

Summary

Conflict between evidences (ta'arudh) means two texts meet in a mutually opposing manner, while taufiq is the effort to show that no real contradiction exists between them. Examples discussed include making the intention for a voluntary fast before the sun passes its zenith, makeup prayers after Fajr and Asr, and whether touching the private part nullifies ablution. For prayers after Fajr and Asr, the texts are reconciled by stating that prayer without a reason is prohibited at those times, while prayers with a cause, such as makeup prayers and the greeting prayer for the mosque, are permitted.

Written in our own words from the original fatwa — check the source before acting on it.