Today's New York Times (p. F11) has a review of a Japanese restaurant called Gari. It describes the food and atmosphere in some detail, but doesn't explain the name, so I will.
/gari/, usually written がり, is the special term used for red pickled ginger when it is served with sushi. The ordinary term is 紅生姜 or in kana べにしょうが /beni sjo:ga/ "red ginger". When you buy pickled ginger in a store the container is labelled 紅生姜, but when you eat it in a sushi shop you call it /gari/.
Ginger isn't the only thing that has a special name in a sushi shop. Green tea is normally called 煎茶 せんちゃ /sentja/, but in a sushi shop the special term 上がり /agari/ is used. Strictly speaking, /agari/ means "freshly brewed green tea", but you hear it mostly in sushi shops. /agari/ is a word with quite a few meanings. The most generic is "rise, slope, ascent", but among other things it is the term used to describe the death of fish and insects.