Obs1. All organisms within a species are different (some faster, larger, darker, having different protein labels on their cells). These differences are termed variability.
Obs2. Differences can be passed to some of your offspring (heritability or genetics). These differences arise through mutation by the way.
Obs3. When times get hard (lack of food, space, etc...) some will die.
Who dies? Those that are least able to "fit" their environment and survive to reproduce. This is called survival of the fittest.
Note: this should have been "survival of the fitness" because evolution favors those that have the most offspring that survive to have offspring of their own.
Because organisms generally reproduce to fill their environment times will almost always get hard. Competition with others of your species is termed intraspecific competition.
All of the above lead to a situtation in which organisms compete for survival against others of their species using their variable and heritable traits. The winners get to reproduce and possibly pass on their traits to the next generation.
A study of the animal world easily provides answers. With deer, for example, the weaker ones are caught and killed by predators such as wolves. In this way the weaker traits are not reproduced. Also, the weaker and the sick will die in the harshness of winter. Only the larger, hardier deer survive in frigid winter conditions in such places as Canada, for instance.
Disease, too, plays a role in natural selection. Those most able to withstand infection thrive while others fall ill and even die. Nowadays, modern medicine often interferes with the process of natural selection as people and animals both survive certain conditions when heretofore they may not have thrived.
In other instances, certain species of animals were so frail that the introduction of other species into their environment resulted in their extinction. A famous example is that of the do-do bird which could not fly, and, thus was vulnerable to any predatory animal introduced into its environment.
Because there is such a delicate balance of Nature, this balance is easily disrupted. In the efforts to control a balance, man has often interfered. However, this intervention has reaped devastating effects at times. A science-fiction author such as Ray Bradbury, in his "A Sound of Thunder" and "Harrison Bergeron" portrays the effects of interfering with nature and natural selection. And, Aldous Huxley's "Brave New World" is a profound examination of what happens when science controls human nature. Of course, the most infamous case of finding a "reason" for selective breeding and the elimination of natural selection in human affairs is in the history of Adolf Hitler.