Gebrochts
Gebrochts (Yiddish געבראָכטס, lit. 'broken', also gebrokts) is a term used to describe matzo that has become wet. (In contemporary Hebrew literature, "gebrochts" is known as "matza sh'ruyah", or "soaked matzo.") Gebrochts are an issue of Passover kashrut, and thus irrelevant the rest of the year. The issue of gebrochts' permissibility during Passover pertains solely to the Ashkenazi community. Non-Ashkenazi Jews do not consider gebrochts to be an issue.
Source
During the holiday of Passover, Jews are forbidden to eat any of five grains specified in the Talmud (namely wheat, barley, spelt, oats, and rye) if they have been "leavened". "Leavening" as it is used here (חמוץ, chimutz, in Hebrew) occurs if flours of these grains are combined with water, and subsequently sit "too long" (generally, more than 18 minutes) before being baked.
Thus, once flour has been reacted with water and rapidly baked into matzo, it is no longer subject to ambient leavening. According to this argument, then, matzo and all of its derivatives are neither "leavened" nor "leavenable", and, therefore are permissible for consumption during Passover. A reading of the tractate Pesachim from the Babylonian Talmud (c. 500) makes it clear that in Talmudic times, matzo soaked in water was permitted during Passover; the Ashkenazi rabbi and exegete, Rashi (c. 1100), also indicates that this was unobjectionable (Berachot 38b).
However, the custom later developed among some Ashkenazim, primarily Hasidic Jews, to avoid putting matzo (or any derivative, such as matzo meal) into water (or any liquid), to avoid the possibility that a clump of flour may still exist in the matzo that was never properly mixed with water and thus is now newly still susceptible to leavening. (This appears, for example, in Shulchan Aruch HaRav, c. 1800.) Therefore, some Jewish communities, especially Hasidic Jews, do not eat matzo ball soup during Passover. Many "non-gebrochts" recipes and products now exist for Passover, generally substituting potato starch for matzo meal, including recent novelties such as kosher-for-Passover, non-Gebruchts, potato starch-based ice cream cones and ice cream sandwiches, and even potato starch-based pizza.
Observance
Today, some observers of this custom ("not eating gebrochts") enforce it to the point that matzo put into liquid is considered potentially bona-fide chametz; therefore, they will not eat off dishes that were used for gebrochts. Today, with this in mind, some hotels and restaurants open during Passover indicate on their menus, "if you would like to add matzo to your chicken soup, please notify the waiter so s/he may provide you with a disposable bowl and spoon."
Others observe the custom in much less stringently, i.e., not eating gebrochts themselves, as part of maintaining the custom, but not otherwise regarding gebrochts as chametz.
Many other variations of this custom exist, including some who allow matzo soaked in fruit juice but not water, and some who allow matzo served with congealed liquids such as butter, cream cheese, or schmaltz. Personal customs generally reflect the norms of one's family and community.
Many Ashkenazi Jews, primarily those descended from non-Hasidic groups (such as Misnagdim), never adopted this custom, and thus consider gebruchts to be a "non-issue". While no one argues that one must consume gebrochts during Passover, many consider gebrochts dishes (matzo ball soup, matzo pizza, for example) to constitute an enjoyable and significant rôle in their Passover experience and thus a way to fulfill the mitzva of being happy on Jewish holidays. In fact, the members of some nineteenth century Lithuanian Jewish communities purposely ate gebrochts to demonstrate the permissibility of this practice.
Given the variety of customs within the Ashkenazic community, it is not uncommon to ask one's friends, "Do you eat gebrochts?" before inviting them for a meal during Passover. Gebruchts can also become a point of familial contention if one's practice is different from one's in-laws.
In Israel, Passover is observed for seven days, as mandated by the Torah; those with the custom of not eating gebrochts generally abstain for all seven days. Outside of Israel, however, an eighth day is observed because of a decree of Rabbinic law. On this eighth (somewhat more lenient in certain regards) day, virtually all communities consider gebrochts to be permitted, even those who adamantly treated gebrochts quite stringently for the first seven days.