"Billy Goat Battle" by Becks;
used with permission under CC BY 2.0
Goats in North America traditionally mate between late August and early January. For male goats (buck or billy goat), the time directly before mating is called a rut which is a time where males are obsessively interested in the female goats (doe or nanny goat). During this time male goats will often fight among each other (seen in the image above) and will even release a smelly musty odor--yuck!--in order to communicate both to other males and interested females!
The average breeding age for male goats is between 8-10 months, and a full-grown, healthy male goat can breed with anywhere between 20-40 does. Since goats breed for such a long period of their lives, breeding happens annually and for a lengthy amount of time: the period begins usually around late-August and extends all the way until early-January; however, many goats even breed out of season.