The Shulhan Aruch (O.H. 583:1), based on the Gemara (Keritut 6a), writes that one should eat certain auspicious foods on Rosh Hashana. These include Rubia (lit. black eyed peas), Karti (lit. leeks), Silka (lit. beets), Tamre (lit. dates) and Kara (lit. gourd). In Morocco, Rubia was commonly understood to mean sesame seeds When each food is eaten, a prayer is recited which invokes an idea that is phonetically or contextually related to that food. For example, when Karti is eaten, one says “May it be Your will that our enemies be destroyed.” “Destroyed” in Hebrew is “Yikartu” which sound like Karti. The Rama (ibid.) adds that there is also a custom to eat an apple dipped in honey and to recite a prayer for a sweet year, and to also eat pomegranate seeds, which signify abundant merits. The common custom, especially among Sephardim, is to have a Seder on the two nights of Rosh Hashana in which these foods are eaten.
Normally, when one eats different types of fruit, there is a prescribed order in which these foods should be eaten. Any of the seven species of the Land of Israel take precedence and should be eaten before a fruit that is not one of the seven species. As such, it would be expected that on the eve of Rosh Hashana, one would start with the dates and then go on to eat the apple. Nevertheless, the Tur (O.H. 583) writes that the custom is to start off the night with the apple. Indeed Rabbi Haim Messas (Leket Hakemach, Resh) confirms that this is the common custom. One way to reconcile this is based on the Ritva (Berachot 40b), who writes that the laws of giving preference to different foods only apply when one has no particular desire for one over the other. When one prefers or has a reason to start with one particular fruit, then that takes precedence. In Kabbalah, there is a concept known as Tapuhin Ila’in Kadishin (lit. “Lofty and holy apples”), which gives deep significance to apples. Thus, although the Ben Ish Hai, Rabbi Shlomo Zalman Auerbach (Halichot Shlomo, Rosh Hashana) and Rabbi Shmuel Kamenetzky (Kovetz Halachot, Yamim Noraim, pg. 865) are of the opinion that dates should still precede the apples, the common custom is for the Seder to begin with apples. Furthermore, honey represents Divine judgment, and by dipping a sweet apple in the honey, it is symbolic of our desire for our judgment to be tempered. In Morocco, some had a custom to dip the apple in sugar rather than honey.
Summary: There are several symbolic foods which should be eaten on Rosh Hashana. The custom is to eat the apple before the date.