for "disagreement". Yiddish
for any type of argument, falling out.
A machkolet falls into two categories:
Machloket L'Shem Shamayim, or "a disagreement for the sake of Heaven" or "an argument to glorify G-d". This is an argument over terms of halacha, or laws, or a discussion over the interpretation of text. A machloket like this takes place between peers who acknowledge and respect each other. This type of machloket is a good machloket because it encourages people to engage their minds in the study of Tanach, Talmud, Gemara, etc. and because the end result honors G-d and is created to honor G-d. The classic example of a machloket l'shem shamayim is the one between Beit Hillel (literally 'the house of Hillel', or Hillel's academy and his student's) and Beit Shammai (likewise) over how the Channukah candles should be lit.
Machloket Shelo L'Shem Shamayim, an "argument not for the sake of heaven". This argument drives people from each other, usually with disastrous effects, sometimes turning people away from G-d. A machloket of this sort takes place between adversaries, or people with little respect for their colleagues. In any case, this type of machloket is seen as unecessary and dangerous to the achdut of B'nei Yisroel, or the Children of Israel. The classic example of machloket shelo l'shem shamayim is the rebellion of Korach and his followers that took place in Bamidbar, or Numbers, perek 16.
How serious is a machloket?
According to Rashi's commentary on Bamidbar 16:27: "Come see how terrible machlokes is. A human court can only punish one who has reached puberty. The Heavenly court can only punish one who has reached the age of twenty. Machlokes kills even nursing infants."
with help from:
breslov.org, ou.org, rabbijablon.com. Paraphrasing from Pirkei Avot, perek chamesh.
and most importantly, my notes from Mrs. Wiggin's Chumash class!