Natural breeding refers to the random mating that occurs between two individuals of the same species.
Conversely, selective breeding has been performed for centuries by humans hoping to produce offspring with desirable traits. Farmers have selectively bred their livestock in the hopes of producing healthier, larger specimens. For example, they might hope to produce a cow that gives a lot of milk or a horse that runs well.
Modern breeds of dogs came into existence by careful selection and breeding. All modern dogs can be traced to a wolf-like ancestor. Selections were made for traits like hunting ability, fighting ability, being a tame companion, and appearance.
Many crops that nourish the planet are the results of selectively breeding plants with traits like resistance to insects or an ability to grow in a drier climate. Larger varieties of fruit have been selected for to provide more food for people. These crops are different than their "wild" cousins that grow in nature.
In selective breeding, there is no guarantee the desired traits will be inherited by the offspring. Due to the processes of crossing over and independent assortment (which occurs during meiosis), unique gametes are produced. These gametes may not have the characteristics desired by the person performing the selective breeding.
I have included a link with details of various types of artificial selection (selective breeding) performed by people to try to "improve" the resulting offspring.