Why are we forbidden to wear leather shoes on Yom Kippur? Is it because leather shoes are a sign of wealth, or comfort; or does it have some thing to do with cruelty to animals?
The Talmud (Yoma 73b) lists the five obligatory "afflictions" (i.e. forbidden pleasures) of Yom Kippur:
Maimonides, in his commentary on the Mishna, summarizes the Talmudic discussion saying:
With regard to the specific prohibition against wearing shoes, while there is some debate on the topic (see Minchas Chinuch 313:14), the dominant opinion is that "shoes" means specifically leather shoes (כל מנעל שאינו של עור לא מיקרי מנעל, see Beis Yosef OC 614), and therefore the prohibition applies only to leather shoes.
For the most part, in virtually all older sources, the discussion ends here, as there was apparently no perceived need to provide a special explanation for the the inclusion of wearing shoes (leather or not) among the required afflictions of Yom Kippur.
Among later sources, however, we do find some additional insights on this topic. Perhaps the most basic explanation is that given by Rabbi Samson Raphael Hirsch in Horeb (in his discussion of the laws of mourning (314:2), which he refers us to in his discussion of the laws of Yom Kippur (158)):
Unfortunately, R' Hirsch does not provide a source for this insight. However, the basic idea, i.e. that shoes symbolize the human capacity for self-sufficient action, can be found in several earlier sources (such as the Abudarham's commentary on the blessing, "שעשה לי כל צרכי").
Like R' Hirsch, a number of other commentaries associate the prohibition against wearing shoes on Yom Kippur with the prohibition against wearing shoes in the Temple, arguing that on Yom Kippur the entire earth is sanctified akin to the Temple and we are therefore required to walk barefoot. (R' Moshe Chagiz, cited in many sources, and R' Menachem Mendel of Rimanov, cited in Agnon's Yomim Noraim and in Artscroll's Yom Kippur)
The Arizal (in Shulchan Aruch HaAri, cited in Taamei Haminhagim) gives an esoteric kabbalistic explanation for why specifically leather shoes are forbidden. The basic point he makes is that leather shoes are associated with the "garments of skin" that God gave Adam and Eve after they sinned (Genesis 3:21) (which, according to medrash, were made from the shed skin of the serpent). Leather shoes therefore symbolize sin and impurity, which have no power on Yom Kippur. Thus, we are required to remove these shoes.
Finally, in a work titled Siddur HaMinhagim written by Rabbi Shlomo Tzvi Shik in the late 1800's, Rabbi Shik gives two original explanations for the prohibition against leather shoes.
The first, based upon a concept found in the commentary of the Shelah Hakadosh on the blessing "שעשה לי כל צרכי", is that leather shoes symbolize man's dominion over all of creation. However, R' Shik argues, a sinner has no such claim of dominion. Thus, on Yom Kippur, when we engage in repentance and confess our sins, we remove our leather shoes to demonstrate our recognition of our sinfulness.
R' Shik's second explanation is based on a custom that when a person would wear a new garment, people would bless him that, "You should wear it out and get a new one." However, some write that one should not say this with regard to a leather garment, as this would require killing an animal and Scripture (Psalms 145:9) states, "His mercy is upon all His creations." Similarly, when slaughtering an animal for the first time, while one does recite the blessings of Shehecheyanu on the mitzva of kisui hadam (covering the blood, assuming it is required, as with a bird), one does not recite Shehecheyanu over the actual slaughtering because, according to some, it involves harming a living creature. Similarly, R' Shik argues, being that Yom Kippur is a day of Divine mercy, it is improper to wear leather shoes.
This latter explanation (which, despite its rather obscure origin, has gained remarkable popularity, mainly because of its inclusion in Agnon's Yomim Noraim (and in the very popular, and heavily abridged, translation, Days of Awe), and, in turn, in many other such collections (including Artscroll's Yom Kippur)) is actually rather difficult in that it would imply that all leather garments should be avoided on Yom Kippur, and there is no such practice (although Agnon does cite a source that the Terumas HaDeshen, R' Yisrael Isserlein, preferred to avoid wearing any leather garments on Yom Kippur).