A durometer is a device for measuring the hardness of a material, as well as the name for the measure of that hardness (aka "Shore A" in the softer ranges). While you can use a durometer to test the relative hardness of metals, in terms of body modification a durometer is usually being used in reference to silicone and other soft materials used for implants. The higher the durometer number, the harder the material.
To put it into context, the O-rings on body jewelry are usually about a 60 or 70 durometer — skateboard wheels are usually between 80 and 100. A very soft squeegee would be about a 55 durometer, and a hard one would be about 80. The softest rubbers are usually about a 20 durometer. Implant silicones come in 10 durometer and up.