A farmer is spraying his crops with a specific pesticide. He notices a drastic decrease in the insect population, so he uses the same pesticide the next year. This time, more insects survive. The next year, almost no insects are killed by the pesticide. Why is this happening?

Probably insects developed resistance to the pesticide.

Explanation:

The term resistance refers to an inheritable change in the population sensitivity, reflected through the consecutive failure of the chemical effects, correctly used in order to reach a certain effect on the insect population.  

Insecticides might produce a genetic modification in the insects, leading them to survive under the effects of the chemical. Insects evolve with the capability of tolerating the poison dose that normally is used to destroy a normal population of plagues.

In the exposed example, the first time that the farmer sprayed the pesticide, the insect population decrease sharply. This means that the chemical affected them severely. The surviving individuals probably suffered a genetic modification and survived. They probably passed this modification to the following generations, which expressed it, and developed resistance to the pesticide.