Darwin's theory of natural selection states those organisms with the best adaptations to the organism's environment have the best chance of surviving. This means the genetic code possessed by those surviving organisms is the code that gets passed along. This is the form of that organism that survived, while other organisms didn't.
A good example of evidence to support the theory of natural selection is the case of the peppered moths in England. Before the industrial revolution, the pale, lighter version of the peppered moth blended in well with it's natural surroundings. The darker version of the peppered moth stood out and was more likely to be eliminated. When the factories in England started covering everything with soot, the existing environment turned from light colored to dark colored. Now the dark version of the peppered moth blended into the environment, while the lighter, pale version stood out. As a result, the dark peppered moth was naturally selected to survive because of it's ability to blend into the environment.
Another good example is the development of antibiotic resistant bacteria. The development of antibiotics has helped save countless lives. Antibiotics are substances that kill harmful bacteria, such as streptococcus bacteria. Over time, some bacteria develop a resistance to the effects of the antibiotic. This bacteria is more likely to survive and pass this ability on to it's offspring. Eventually, the antibiotic becomes less effective, due to the bacteria's development of a resistance to the substance in the medication.