When taken in the evolutionary context, "survival of the fittest" has a much different meaning than most people apply to it. "Survival" in this sense does not mean the perserverance of the ogranism, but rather is in reference to the continued lineage of the organism's genes. "Fittest" has nothing to do with physical health or strength, but rather the ability to best adapt to the most effective reproductive practices.
As a perfect example of these context specific meanings, take the male partner to the black widow spider. The male black widow is very small compared to its female counterpart and is not nearly as strong. Each individual male also has a built in mating ritual that is catastrophic for its own continued existence, but allows its genes to survive and even thrive in the next generation. After successfully impregnating the female, the male black widow will sacrifice itself as food for the female. The female uses the nourishment provided by her mate's corpse to remain strong and healthy and provide needed energy to ensure the growth of the couple's offspring. The male's progeny benefit not only from his genes but from his donation of caloric energy.
Genes and behaviors are affected by this "survival" rule. From the size of an organism's gametes (reproductive cells) to the energy it expends to care for its young, almost anything about an organism's genotype or behavior can be attributed to its need to successfully pass its genes along into the future.
The very division of many lifeforms into male and female sexes shows the importance of natural selection and benefits of sexual reproduction. Males have developed smaller gametes (sperm) than females (ova) because males have evolved to put comparitively little energy into each reproductive opportunity but to take as many reproductive opportunities as possible. Males produce trillions of sperm cells and females only release a comparitively small number of ova. Females make a greater energy sacrifice in both the production of their larger gametes and the more devoted care to their young. Females put a large amount of energy into each reproductive opportunity. One sex has chosen quality, the other quantity. This differentiation is beneficial for the entire species.
Sexual reproduction, which mixes genes equally from two parent organisms to create an offspring, has many evolutionary advantages over its alternative, asexual reproduction. Asexual reproduction allows an organism to create exact clones of itself, which on first inspection would seem the perfect way to pass on genes. However, as a community these clones all share the same weaknesses and vulnerabilities, a single disease could wipe out the entire population or a single mutation could perpetuate throughout the population and cripple future generations. Sexual reproduction has a greater efficacy in allowing genes to survive because the genes of one sexual partner can provide defenses that cover the vulnerabilities and mutations of the other.
Natural selection has ensured that the genes and behaviors that pass on within a certain species are those that make the organisms the "fittest" they can be and allow for the "survival" of the species and of life itself.