Chametz ("leaven") is anything made from one of the five grains (wheat, barley, spelt, rye, oats) where the unbaked grain has come into contact with water for more than 18 minutes. When you make dough for bread, cake, pizza, or noodles, the flour touches water before baking for a much longer period – so all of this stuff is chametz. To bake the matza (unleavened bread) used on Pesach, care is taken that from the moment the flour touches water until baking is finished no more than 18 minutes elapss [there is also year-round matza where no such care is taken, this renders the matza chametz, so take care that the matza you buy for the holiday has "kosher lePesach" (fit for use on Pesach) written somewhere.].
So, it is universally accepted that the leavening process starts after exactly 18 minutes. Why 18 minutes? Where does this number come from?
Said R. Abbahu in the name of R. Simeon b. Lakish: [The period for fermentation is] as long as it takes a man to walk from the Fish Tower [Migdal Nunia] to Tiberias, which is a mil. Then let him say a mil? – He informs us this, that the standard of a mil is as that from Migdal Nunia to Tiberias.
This has been codified in halacha, e.g., in the Shulchan Aruch Orach Chaim 459:2 or by Rabbi Moshe ben Maimon (the Rambam) in the Mishne Tora, Laws of Chametz 5:13 (English translation from Chabad):
[if] the dough has lain at rest for the time it takes a man to walk a mil, it has become chametz and must be burned immediately.
A "mil" is dfined as "two thousand cubits" or roughly a kilometer. According to most opinions this amounts to an 18-24 minutes walk. For baking matza, we are strict and take the shorter time, i.e., 18 minutes, in order to be on the safe side.
The number 18 is also the numerical value of the word "chai" (Hebrew for life). Some people read nice explanations into the fact that after 18 minutes the dough comes "alive", but I couldn’t find anything about that on a reliable web page, so I doubt the connection.
(c) Credits go to Double AA at Mi yodea.