Survival of the fittest was first proposed by Herbert Spencer,and not Charles Darwin as many people think, and is a relatively simple term which explains the process of evolution to a degree. The phrase itself basically means that whoever is best suited to survive within any given situation will do so, although can only be applied to the immediate situation, and not indefinite survival of a particular species. It is also not used very often by scientists these days as it over simplifies the process of evolution, and doesn't actually explain survival beyond the immediate life of a single organism.
Many people tend to use survival of the fittest when referring solely to either physical strength of an individual and their likelihood to mate. Of course this idea is flawed as other traits are often a lot more important when determining a species ability to survive than prowess of individuals and their offspring. If this were the only factor, then populations would quickly become stagnated and weak as inbreeding within a small group of genes would occur. For example despite a large number of individuals surviving, the ability to attract a mate is often more important than simply surviving.
Despite the fact that survival of the fittest is often used in conjunction with natural selection, the preferred and more widely accepted term, they are in fact different. Confusion between the two terms can occur in a situation such as the fact that male peacocks attract mates, and are therefore more likely to reproduce, by having the largest plumage. At the same time however the fact that they have larger plumage might make them slower, and therefore more likely to be both seen and caught by predators. So although this situation is a process selection, it isn't necessarily applicable to survival of the fittest, as the fittest would technically be those individuals who had smaller plumage and could run faster.
This is because survival of the fittest should only be used for particular situations, and not to the wider survival of a species. Natural selection on the other hand refers more to inherited traits which make an individual more or less likely to be able to survive and mate, passing on its traits to another generation. So a bird with large inherited plumage is more likely to be naturally selected, but at the same time is less likely to survive an encounter with a predator.
Other factors also affect natural selection, that can drastically alter the paradigm of what the fittest might be aren't catered for in survival of the fittest, as what makes something the fittest can change. For example having a particular gene mutation in order so survive a deadly virus could become the factor in determining both the fittest individuals, as well as those naturally selected. Also in the true sense of the process of evolution, it is not only the strongest or fittest that survive.
In actual fact a more accurate idea of survival of the fittest should be survival of the fit enough, otherwise of course every species would progressively bet stronger and bigger with each generation. Of course this doesn't happen, and in actual fact a larger percentage of the population of any given species survives than simply the very fittest.