There is a custom of eating an apple dipped in honey on first night of Rosh HaShanah (among the other foods) as a sign that we should have a sweet year. Why is this done specifically with an apple and not another fruit?
1) The Maharil explains that the apple is connected with "חקל תפוחים קדישין"; when Yaakov came to get the brochos from Yitzchok, he had the smell of an apple orchard upon his clothing. Besides for the Kabbilistic meanings, (according to one opinion) this episode happened on Rosh Hashana (GR”A O.C. 583:8)
2) There are three types of benefit derived from an apple; taste, sight and smell. This is symbolic of our appeal for sustenance in the three aspects of children, health and livelihood for the entire year (Ben Ish Chai, first year Parshas Nitzavim)
3) It is written in Zohar (Parshas Shmini 4a) that after one drinks wine, they eat an apple in order that the wine should not harm them. Wine represents gevurah (severity) and thus the apple calms the severity. Therefore on Rosh Hashana we eat apples to pacify the harsh judgement. (Ben Ish Chai there)
4) The Zohar (there) states that source of all fruit is from the Divine aspect of malchus (feminine), besides for the apple which is sourced the aspect of Tiferes (male). The Arizal says that the summer months represent malchus and the winter months (Tishrei through to Nissan) are Tifferes. Therefore at the start of the “male” months we eat an apple which comes from “male” source. (Ben Ish Chai there)
5) According to Rabbeynu Yona (brought in Rosh Brachos Perek 6 Siman 35) that the nature of honey is to change everything that comes into it – even something impure – into honey. (Therefore one can comsume honey even though the legs of the bees may be mixed in). The holiness of Rosh Hashana should convert all bad to good. (Shem Mishmuel Mo'adim Rosh Hashana 5674) [Some add that the sin of Adam Harishon and the Tree of Knowledge was with an apple, and we therefore dip an apple into the honey to rectify the original sin].
6) The Gematria of "תפוח" is equivalent to that of "פרו ורבו", and "דבש" is equal to "אשה", symbolizing that Rosh Hashana is an auspicious time for barren women to be remembered (Imrei Noam Chelek Beis, Likutim Besof Hasefer Ois 9)
Taamei HaMinhagim 706 says it is done for Kabalistic reasons. In the notes, he mentions in the name of the Imrei Noam that the Gematria of the word "Tapuach" (the Hebrew word for apple) is the same as the Gematria of "S'e Akeida" - so we eat the apple to recall the Akeida (Binding of Isaac).
There is specific symbolism in the apple. It also can't be discounted that apples are harvested around Rosh Hashana time so they are a readily available and relatively inexpensive fruit at that time.
Apple was the most widely known fruit. Think of the other fruits and vegetables that are called after the apple eg orange in Hebrew, potatoe in Hebrew, French and German, pineapple in English. See here "In Middle English and as late as 17c., it was a generic term for all fruit other than berries but including nuts (e.g. O.E. fingeræppla "dates," lit. "finger-apples;" M.E. appel of paradis "banana," c.1400)."
two chiddushim of my own that I have not seen elsewhere.
the pyre on the mizbeach is described as an apple. We are adjured not to offer up honey on the mizbeach because no man can stand it, but l’atid lavo we will be able to offer up honey on the mizbeach - so our tefillah is that the year should be sweet enough for us to do so and thus we dip the apple in honey
The jewish people are compared to an apple (k’tapuach b’atzei hayar) - (the academic interpretation non-withstanding) Yerushalayim is compared to a barrel of honey by our enemies (gittin 59b?) we learn in krisus that simanim milsa hi (6b?) we also learn that the goyim have their own simanim (nasu ososam osos (also gittin 59b) we are trying to be makdim our sign to theirs (like our shekalim before haman’s) and we dip the apple in honey as a tefillah that klall yisroel should be reunited with a rebuilt yerushalayim