The way I see it, the relationship between the two words is simply explained like this: Extinction is what happens to species that can't cut the test of natural selection.
Charles Darwin is the man credited with popularising the term "natural selection" in his book The Origin of Species.
Extinction and natural selection
Natural selection is the cornerstone of evolutionary theory and is the antonym for what Darwin described as artificial selection (or what we now call "selective breeding").
Darwin believed that all life on Earth was constantly evolving, and this evolution was due to mutations at a genetic level. Whilst mutations can occur fairly frequently within a species (for example dimples are a genetic mutation), the mutations that affect evolution are far rarer.
The mutations that impact evolution are the ones that help the mutant offspring survive everything that nature and its environment throws at it, where the non-mutated ones die out (become extinct) instead.
This process is called natural selection, because nature is effectively choosing who survives and who doesn't!