Why does hydrogen sulfide smell like rotten eggs?

Hydrogen sulfide ($H_{2}S$) is a colorless, extremely poisonous gas that is naturally produced by the decay of organic substances containing sulfur. This process of putrefaction releases the gas, which is responsible for the characteristic odor often described as smelling like rotten eggs.123

In addition to its presence in decaying matter, hydrogen sulfide is frequently found in volcanic vapors and mineral waters. Because it is a byproduct of the breakdown of sulfur-bearing organic materials, its presence serves as a common indicator of such decomposition.1