The definition
An amuse-bouche is a single bite the kitchen sends to each guest before the meal, not ordered and not charged for.
It is a greeting from the kitchen and a preview of its style: a spoon of something cold, a tiny tart, a cup of soup. It differs from an appetizer on both counts: an appetizer is chosen by the guest and priced on the menu.
How you hear it
Guests who have not seen one before sometimes worry it will show up on the check. Saying “from the kitchen” and “on the house” settles it.
Where it shows up on the menu or the check
On a tasting menu, the amuse-bouche is often the first line, set in lighter type and without a price, since it is part of the meal. On an à la carte menu it usually does not appear at all; the server announces it.
On the check, it never appears. If the kitchen sends extra courses beyond the amuse, those are usually rung in and comped so the kitchen can track what went out.
Some kitchens change the amuse daily from trim and leftover prep, which is one reason it rarely gets printed: it may not be the same tomorrow.
What it means for prices and printing
An amuse costs the kitchen food and time on every cover, so it lives in the budget of the meal around it, usually a tasting menu or a higher check average room. Keep it small and made from what the kitchen already has.
In Gaslamp, every item needs a price of at least $1, so an unpriced amuse cannot be its own line. List it in the description of the tasting menu line (“amuse-bouche, then six courses”) or in the note under the first course heading. On an à la carte menu, leave it off and let the server announce it.
Allergies are the one place an amuse needs planning. It arrives before the server has asked about allergies at many tables, so the safe order is to ask first and send it second, and to have a second version ready for the guest who cannot eat the first.
Where it comes from
Wiktionary gives it as borrowed from French amuse-bouche, from amuse plus bouche (mouth), a politer form of the older amuse-gueule, where gueule is the slang word for mouth.
Go deeper: Prix fixe and tasting menus