A movement among German phoneticians in the 1870s, whose tenet was that all sound changes follow a rule. There are no sporadic exceptions, and every apparent exception to a law of sound change can be explained by finding some other law, or some other good reason why a different form was adopted. This is called the Regularity Principle.

The Neogrammarian hypothesis was controversial in its time, and underwent theoretical challenges over the next half century, but it has long been accepted by almost all linguists. It may therefore be called an axiom not only of phonetics but of linguistics: there is always a good reason for any change, and it can in principle be found.

The original German term is Junggrammatiker. It was, I gather, first applied to the Neogrammarians by their opponents. The principle was explicitly stated by August Leskien in 1876. Another prominent Neogrammarian was Karl Brugmann, and their major text was Hermann Paul's 1880 Prinzipien der Sprachgeschichte (Principles of Language History).

At that time the most impressive example of a sound law was Grimm's Law, which explained the consonants of the primitive Germanic languages in comparison with the consonants of Latin, Greek, and Sanskrit.

The wide application of Grimm's Law to a whole range of consonants was marred by exceptions in a smaller number of words. The Danish linguist Karl Verner put forward an explanation of these in 1876, showing that developments in Germanic went in a different but consistent direction when the accent was on the following syllable.

For example, Grimm's Law predicts the following matches:

Latin   Greek   Sanskrit   Proto-Germanic   Proto-Indo-European
  p       p        p            f                p
  t       t        t            th               t
  f       ph       bh           b                bh
The Proto-Germanic is the reconstructed (not known from writing) ancestor of English, German, Scandinavian, and so on, and the Proto-Indo-European (= PIE) is the reconstructed common ancestor of all four of the others.

So Latin frater, Greek phrater, Sanskrit bhratar, and Germanic brother all match. (The vowels are also related by known laws.) If we just use modern English father, then that too matches Latin pater, Greek pater-, and Sanskrit pitar, but this shows the perils of incomplete data. This would not look like an exception if modern English was all we had. But in Old English it was fader and in German it's Vater (where V = F). These both imply an original Germanic fader, which fails to correspond by Grimm's Law.

However, given that the Proto-Indo-European forms were accented bhráter but patér, Verner's law applies to the second, and all the correspondences are once more regular.

Lack of data is one reason why apparent exceptions continue to be problematic. Languages go extinct and information is lost. Another problem is that subsequent changes can overlay the results of former ones. The English and German words for father illustrate this. In German a second sound shift applied in the early Middle Ages, like a repeated application of Grimm's Law. Water became Wasser and Vader became Vater (at the same time - the t > ss change didn't then apply to the new t arising from d.)

The English illustrates a different subverter of regularity: analogy. The ancestral PIE word máter became mother in Old English, so they had fader, mother, brother. Analogy is a well-known influence that causes forms to change. The change from fader to modern father might have been because of some later sound law in Middle English, but I don't know of any that would do this. Levelling by analogy is the simplest explanation.

In the early 1900s the Neogrammarian movement was opposed by a group called Neolinguists in Italy. They believed that individual mutations and idiosyncrasies could become established without an underlying general rule. They derived some support from the detailed linguistic atlas of France created by Jules Gilliéron, which showed wide varieties of word forms from region to region. However, Gilliéron was neogrammarian in explaining his forms. In 1953 the great linguist Joseph Greenberg could declare that the neolinguistic ideas were not generally accepted.