Method 1: Through Gluttony
There used to be hundreds of thousands of giant tortoises roaming (slowly) about South America's Galapagos Islands. Today, there are roughly 15,000. What can we say? Turtles are tasty. In the 16th and 17th centuries, the Galapagos were the swashbuckler's equivalent of a 7-Eleven--the last chance to stock up on food before hitting the vast emptiness of the Pacific. Besides being sluggish and docile (i.e., easy to catch) the tortoises could also survive for up to a year without food or water. Sailors often captured hundreds at a time, stacked them on their backs and, thus, had fresh meat all the way to India.
Method 2: Out of Sheer Hatred
Passenger pigeons once traveled around the United States in flocks so large, they could reportedly block out the sun over a town for eight hours. In the process, they gobbled down all the fruits and grains they could get their beaks on and left the "remains" for farmers to step in. All this made them rather ... unpopular. Throughout the 19th century, killing passenger pigeons was basically the national pastime. Baited with alcohol-soaked grain, gassed with sulfur fires and loaded live into trapshooting launchers (they were later replaced with clay "pigeons"), the passenger pigeon population quickly petered out. The last one died in the Cincinnati Zoo on Sept. 1, 1914.
Method 3: Via Tragicomic Irony
Collector and proto-environmentalist Rollo Beck visited the island of Guadalupe, off Baja California, on December 1, 1900. During the trip, he sighted a flock of nine Caracaras, a rare bird he wished to study (apparently in taxidermied form), and so he shot down all but two of them. Those two turned out to be the last Caracaras ever seen alive.
It's really sad about the Caracaras, but the passenger pigeons kind of had it coming.