Rosh Hashanah would not be the same without dipping the apple into honey or some other sweet substance like sugar. The Talmud Shabbat 88a compares the Jewish people to an apple tree in the orchard. This, say the sages, teaches you that just as an apple tree starts to bear fruit before its leaves grow, so too did the Jewish people put ‘We will do’ before ‘We will hear' at Mount Sinai. So apples have a quality in their development that they are willing to grow without the protection of the blossom, we as a nation have a similar quality, we push ourselves to grow, even if we are not certain of our spiritual success.
When Yitzchak blesses Yaakov he references this Jewish nation and apple connection by telling him "Behold, the smell of my son is like the smell of a field which the Lord has blessed” (Bereishit 27:27). This field, says Rashi, was an apple orchard. When Yitzchak smelled the scent of an apple orchard, he realized that Yaakov’s children were truly worthy of blessing. He saw in the sweet scent of apples an omen that even when Yaakov’s children become entrenched in sin, they have the ability to swiftly extract themselves from their plight. Love of God was so much a part of Yaakov that he passed that love along to his children as an almost hereditary trait. This ability to love God and return to Him from any distance is represented by the scent of an apple orchard.
So the custom is to dip that apple, i.e. the Jewish people, into something sweet, such as honey or sugar (I am not sure if artificial sweeteners work for this, we want our sweetness to be real!) asking God to allow His people to have a sweet year of only goodness.