Everything we do on Rosh Ha-shana has implications for the whole year. Since this is the first day of the year, when life is allotted to every being, every action, word, and thought of the day impacts upon the whole year. This is the meaning of our Sages’ principle, “Symbols are a real thing” (Keritot 6a). That is, symbolic acts have significance; if one performs a symbolic act of blessing at the beginning of the year, it will lead, we hope, to blessing throughout the year.
Based on this, the Gemara recommends that at the Rosh Ha-shana meal, people eat foods that have auspicious implications for the whole year: karti (leek), so that “our enemies may be cut down (yikartu)”; rubya (black-eyed peas), so that “our merits may be plentiful (yirbu)”; tamar (dates), to signify that “our enemies and sins may come to an end (yitamu)”; selek (beet), “so that our enemies may be removed (yistalku)”; and dela’at (pumpkin), which symbolizes blessing, as it is large and fast-growing (SA 583:1). It is also customary to eat apples dipped in honey or sugar water, to symbolize a sweet and good new year, and pomegranates, whose many seeds symbolize our wish for our merits to increase (Rema ad loc.). Many have the custom to eat from the head of a ram or a fish, symbolizing our wish to be the head, not the tail (SA 583:2). It is also customary to eat fish, as they are very fertile and not subject to the evil eye.
Based on this principle, there are additional customs to eat a variety of foods whose names, shapes, or tastes are positive omens for the new year; each community’s customs are based on its languages and the foods that were available to it. When possible, it is good to carry on these traditions.
We do not just eat these foods. Rather, Rishonim say that it is proper to add a short prayer (Yehi Ratzon) before the consumption of each food. For example, before eating dates one should recite: “May it be Your will, Lord our God and God of our fathers, that our enemies come to an end.” Shlah explains that the primary goal of all of these symbolic foods is to inspire us to pray and repent. Every prayer on Rosh Ha-shana has a serious impact on what happens during the course of the year. Therefore, it is proper on Rosh Ha-shana that even eating itself be permeated with prayers for the upcoming year to be good and sweet (Shlah, Masekhet Rosh Ha-shana, Ner Mitzva §21).
Many customarily dip the challah on which they recite Ha-motzi in honey or sugar to symbolize our wish for good and sweet year (MB 583:3). Some dip the challah into salt first, but use only a little, so that it will not impair the sweet taste. Others simply leave salt on the table but do not dip the bread in it. Some keep dipping challah in honey until Simḥat Torah. Others do so only on Rosh Ha-shana. All of the customs are legitimate, and everyone should continue with his family custom.
It is customary on Rosh Ha-shana to eat good, fine foods as a good omen for the entire year. Many also avoid eating unripe fruit or preparing sour, salty, or bitter foods (Rema 583:1; MB ad loc. 5). This is the custom of Ashkenazim and many Sephardim (Ḥida; R. Ḥayim Palachi; Ben Ish Ḥai; Kaf Ha-ḥayim 583:18). Nevertheless, many people do eat sharp foods, although since many Ashkenazim tend to eat sweet foods, they are anyway not eating sharp, peppery foods.
There is an Ashkenazic custom to avoid eating nuts on Rosh Ha-shana, because they allude to sin and because they can increase phlegm and sputum and lead to disturbance of the prayer service (Rema 583:2).
One who does not want to eat one of the symbolic foods, for example because he does not like the taste, may simply look at it. This is because some maintain (based on the wording of the Gemara in Horayot 12a) that the symbolic foods are meant to be pointed at rather than eaten.