Brugmann's law, named for Karl Brugmann, states that Proto-Indo-European (the ablaut alternant of *e) in non-final syllables became *ā in open syllables (syllables ending in a single consonant followed by a vowel) in Indo-Iranian. Everywhere else the outcome was *ǎ, the same as the reflexes of PIE *e and *a. The rule seems not to apply to "non-apophonic *o", that is, *o that has no alternant, as in *poti- "master, lord" (thus Sanskrit pati-, not *pāti (there being no such root as *pet- "rule, dominate")). Similarly the form traditionally reconstructed as *owis "sheep" (Sanskrit ǎvi-), which is a good candidate for re-reconstructing as *h₃ewi- with an o-coloring laryngeal rather than an ablauting o-grade.
The theory accounts for a number of otherwise very puzzling facts. Sanskrit has pitaras, mātaras, bhrātaras for "fathers, mothers, brothers" but svasāras for "sisters", a fact neatly explained by the traditional reconstruction of the stems as *-ter- for "father, mother, brother" but *swesor- for "sister" (cf. Latin pater, māter, frāter but soror; note, though, that in all four cases the Latin vowel in the final syllable was originally long). Similarly, the great majority of n-stem nouns in Indic have a long stem-vowel, such as brahmāṇas "Brahmins", śvānas "dogs" from *k'wones, correlating with information from other Indo-European languages that these were actually on-stems. But there is one noun, ukṣan- "ox", that in the Rigveda shows forms like ukṣǎṇas "oxen". These were later replaced by "regular" formations (ukṣāṇas and so on, some as early as the Rigveda itself), but the notion that this might be an *en-stem is supported by the unique morphology of the Germanic forms, e.g. Old English oxa nom.singular "ox", exen plural—the Old English plural stem (e.g., the nominative) continuing Proto-Germanic *uχsiniz < *uχseniz, with two layers of umlaut. As in Indic, this is the only certain Old English n-stem that points to *en-vocalism rather than *on-vocalism. (Some additional jickering is necessary to account for the Old English details, but nothing serious.)
Perhaps the most startling confirmation comes from the inflection of the perfect tense, wherein a Sanskrit root like sad- "sit" has sasada for "I sat" and sasāda for "he, she, it sat". It was tempting to see this as some kind of "therapeutic" reaction to the falling-together of the endings *-a "I" and *-e "he/she/it" as -a, but it was troubling that the distinction was found exclusively in roots that ended with a single consonant. That is, dadarśa "saw" is both first and third person singular, even though a form like *dadārśa is perfectly acceptable in terms of Sanskrit syllable structure. This mystery was solved when the ending of the perfect in the first person singular was reanalyzed as *-h₂e, that is, beginning with an a-coloring laryngeal: that is, at the time Brugmann's Law was operative, a form of the type *se-sod-h₂e in the first person did not have an open root syllable. A problem (minor) for this interpretation is that roots that pretty plainly must have ended in a consonant cluster including a laryngeal, such as jan- < *ǵenh₁- "beget", and which therefore should have had a short vowel throughout (like darś- "see" < *dorḱ-), nevertheless show the same patterning as sad-: jajana 1sg., jajāna 3sg. Whether this is a catastrophic failure of the theory is a matter of taste, but after all, those who think the pattern seen in roots like sad- have a morphological, not a phonological, origin, have their own headaches, such as the total failure of this "morphological" development to include roots ending in two consonants. And such an argument would in any case cut the ground out from under the neat distributions seen in the kinship terms, the special behavior of "ox", and so on.
Perhaps the most worrisome data are adverbs like Sankrit prati, Greek pros (< *proti) (meaning "motion from or to a place or location at a place", depending on the case of the noun it governs) and some other forms, all of which appear to have ablauting vowels. They also all have a voiceless stop after the vowel, which may or may not be significant. And for all its charms, Brugmann's Law has few supporters nowadays (even Brugmann himself eventually gave up on it, and Jerzy Kuryłowicz, the author of the brilliant insight into the sasada/sasāda matter, eventually abandoned his analysis in favor of an untenable appeal to the agency of marked vs unmarked morphological categories. Untenable because, for example, it's a commonplace of structural analysis that 3rd person singular forms are about as "unmarked" as a verb form can be, but in Indic it is the one that "gets" the long vowel, which by the rules of the game is the marked member of the long/short opposition).
Brugmann, Karl (1876). 'Zur Geschichte der stammabstufenden Declinationen, Erste Abhandlung: Die Nomina auf -ar- und -tar-'. Curtius Studien, 9: 361-406.
Hirt, H. (1913). 'Fragen des Vokalismus und der Stammbildung im Indogermanischen'. IF, 32: 236-247.
Jamison, Stephanie (1983). Function and Form in the -aya- Formations of the Rig Veda and Atharva Veda. Göttingen.
Lubotsky, Alexander (1990). 'La loi de Brugmann et *H3e. La reconstruction des laryngales'. Bibliotheque de Faculté de Philosophie et Lettres de l'Université de Liège, fascicule CCLiII. Liège-Paris: 129-136
---(1997). 'Review of: Marianne Volkart, Zu Brugmanns Gesetz im Altindischen. (Universität Bern, Institut für Sprachwissenschaft. Arbeitspapier 33.) Bern 1994.