Particularly annoying little seed-containing thorns that are shaped like the head of a goat (hence the name). They often have multiple, very sharp points, although each thorn generally has one large point that stands out compared to the others. Generally no matter which way a goathead falls on the ground, there will be a point sticking up.
Goatheads are the bane of cyclists in many areas, especially in the southwestern United States. In fact, the only people that enjoy the existence of goatheads are the makers of puncture-resistant tubes and tires for bicycles, such as Turbo Armadillo tires and Slime tubes. These downright aggressive thorns can easily puncture all but the most durable (and expensive) tires. Even costly Kevlar-weave tires can fall victim to these vicious little beasts.
The thorns grow on ground-hugging plants that spread rather rapidly across the landscape, and each plant can be the source of dozens of goathead thorns. Goathead plants seem to bloom multiple times throughout the year, bringing fresh crops of goatheads every few months. The thorns can travel some distance on wind, the bottom of shoes, etc. thus helping the seeds spread far and wide.
The following URL should give you an idea of what goatheads look like: http://www.goatheads.com/goathead.jpg