Asafetida gets a bad rap from some, and is highly touted by others, and a clear accounting of something as subjective as a flavor is hard to give. But perhaps we can do a bit better than the entries above.
Asafetida smells like you took a clove of garlic, a slice of onion, and a slice of another onion that has been in the compost for two weeks, and mixed them together. This improves with cooking, but the first wave of scent may trigger all of your "Warning -- Rotten Food!" alarms, and may put you off the spice for good. After cooking, the sulfurous, rotting tang is still there, just more mild. Just as humans tune out boar taint, sulfurous eggs, the fermented goodness of wine, kimchi, and vinegar, asafetida also has enough goodness to convince your brain stem that it's good eating. You get used to it very quickly... or you don't, and you hate it forever.