Whoever decided to find out what asafetida tastes like must have been very, very hungry. One of its nicknames - devil's dung - gives an idea of its stinking pungency. It comes from the root of the asafetida plant (whose stalks and foliage look somewhat like fennel fronds) and the thick liquid made from it is dried into a hard mass, which is then crushed or grated into a powder.
Asafetida is used most frequently in the cuisines of India, where it's blended with other spices. When cooked, the pungency becomes mellow and agreeable. Its use is as much for its medicinal effects as for its flavour: it's said to be anti-bacterial, aid digestion, and work as a sedative; while its anti-flatulent properties mean that in Indian cuisine it's usually cooked with beans and legumes.
A little asafetida goes a long way, especially when it's in its lump form. It's often sold crushed and mixed with other ingredients (including starch) to make it easier to use. Look for it in shops that specialise in Indian products. Whether in lump form or a powder, asafetida should be stored in a glass jar with a tight-fitting lid, to contain the smell.