There is a lot of confusion with survival of the fittest. Many people use this as a word that is interchangeable with evolution. In reality though survival of the fittest is completely different then survival of the fittest. Evolution is the process of change over time to become a better species, where as survival of the fittest is the method in which this evolution takes place.
O.K., now that we have some of the confusion out of the way we can describe what survival of the fittest actually is. Survival of the Fittest is a rather simple topic. This topic basically says that the animals that have disirable traits, such as running fast to get away from predators easier, will survive and reproduce, while the animals with less desirable traits will eventually die out and no long give their less desirable traits to the next generation. This is a rather simple topic that just about everyone agrees with.
Now there are some problems with this theory. Many people say that the human race has ultimately overcome survival of the fittest but really you can never get away from survival of the fittest. Sure we may have no natural predators but we are naturally selected in other ways. One of the main ways is through microbacteria. The people who are more resilient to these bacteria are the ones that survive, while the ones that are less resistant to microbateria are more likely to die at a younger age then those that have more resistance. Unfortunately people are dieing because of this natural selection, but unfortunately that is how it works.
Now there are a lot of people that tie this into evolution, mostly because Charles Darwin used this as his original way of passed down traits. Now in his studies he saw many different kinds of finches all within the same island strain. He latter realized that the finches seem to have adapted to the conditions on their particular island. This discovery revolutionized the use of natural selection. Originally natural selection was thought to be how the species evolved in a steady way which came in parallel lines. The discovery of Darwin latter made the flow charts of animals to be more like a tree where natural selection could work in different ways if a particular species was separated in some way. Then became the basis that Darwin set his evolution to. He said basically because of natural selection that evolution is in existence. This wasn't a new topic, but Darwin was the first person to really show that natural selection really does exist, with enough detail in his journals so that everyone could see what he saw. Now people see his work in natural selection every time they read about evolution, the problem is that most people don't really divide the two and see that they do interlace but they are separate ideas, and in theory if one wasn't true the other still could hold up itself.
All in all survival of the fittest is just the process of animals that are strong surviving, while the ones that are weak die. Many scientists have pull this theory way out of proportion, and honestly I hope that you can see that it isn't that complicated as they want you to believe it is, it's a time old theory that is probably going to be around forever.