The main comb shapes are:
There are different variations of the main shapes, usually caused by crossbreeding.
Although the purpose of the comb isn't fully known, there are widely accepted theories on it's use. Seemingly, the biggest use of the comb is as a cooling system for the chicken. As blood flows to the comb, it cools off, as the vessels are closer to the skin, and not insulated by feathers. When the blood returns to the rest of the body, it is cooler than the rest of the blood, and thus helps cool off the body. The same thing can be said of the wattles. As chickens are descended from the red jungle fowl however - a warm climate species - big combs are better suited for chickens in hot climates. In cold climates, they can be a serious threat to the chicken, as they will get frostbite easier than any other part of the body. For this reason, chickens in cold weather climates must be protected from freezing temperatures, especially at night, and if they have large combs. Sometimes the extreme measure of dubbing is used, wherein the comb is partially or completely cut off. Dubbing is sometimes done to prevent frostbite, but often is a requirement for some breeds of chickens for showing. Dubbing has been deemed cruel and is banned in some areas.
It's also believed that the comb could serve as some kind of courtship display, although this has not been proven. Although hens don't seem to prefer roosters with larger combs, the comb can be a display of health, as well as social status. Comb size and hue are influenced by sex hormone levels which is often an indicator of social status.  The size and color of the comb and wattles is also associated with gonad development and the secretion of sex hormones.
A bleached or dark purple comb can be a sign of health problems (except in breeds like the silkie, where purple combs are normal). In hens, the comb can be connected to egg production. For example, a hen's comb will often flop over if she's laying regularly.
As experiments indicate, chickens recogize each other by looking at the head and facial features, and the comb may play a big role in this as well. 
The Comb-Gene connectionEdit
In the first decade of the twentieth century, British geneticists William Bateson and R. C. Punnett conducted research showing that the shape of the comb in chickens was caused by the interaction between two different genes. They were aware of the fact that different varieties of chickens possess distinctive combs. When Bateson and Punnett crossed a Wynadotte chicken with a Brahma chicken, all of the F1 progeny had a new type of comb, which the duo termed a "walnut" comb. In this case, neither the rose comb of the Wyandotte nor the pea comb of the Brahma appeared to be dominant, because the F1 offspring had their own unique phenotype. Moreover, when two of these F1 progeny were crossed with each other, some of the members of the resulting F2 generation had walnut combs, some had rose combs, some had pea combs, and some had a single comb, like that seen in Leghorns (Figure 1). Because the four comb shapes appeared in a 9:3:3:1 ratio (i.e., nine walnut chickens per every three rose chickens per every three pea chickens per every one single-comb chicken), it seemed that two different genes must play a role in comb shape.
Through continued research, Bateson and Punnett deduced that Wyandotte (rose-combed) chickens must have the genotype RRpp, while Brahma chickens must have the genotype rrPP. A cross between a Wyandotte and a Brahma would yield offspring that all had the RrPp genotype, which manifested as the walnut-comb phenotype. Indeed, any chicken with at least one rose-comb allele (R) and one pea-comb allele (P) would have a walnut comb. Thus, when two F1 walnut chickens were crossed, the resulting F2 generation would yield rose-comb chickens (R_pp), pea-comb chickens (rrP_), and walnut-comb chickens (R_P_), as well as chickens with a new, fourth phenotype—the single-comb phenotype. Based on the process of elimination, it could be assumed that these single-comb chickens had the rrpp genotype (Bateson & Punnett, 1905; 1906; 1908).