Evolution is a process in which life is shaped by the environment it lives in. All living things have genes, which act as their blueprints. Over time, genes spontaneously mutate, causing changes in individual organisms. If those changes allow the organism to bear more offspring than others of its species and the mutations happen in genes which are passed on, then more and more organisms will start to bear those genes over time. As many generations pass, the organisms might become entirely different from what they once were.
As an example, consider an animal which obtains a dark coloring as a result of a mutation. If it lives in a dark environment, potential predators will have a harder time spotting it than they have spotting its lighter-colored mates. Over time, there will be more and more dark-colored animals as the light-colored ones get eaten more often. An organism's capability to bear healthy offspring is called its fitness: an organism that has more offspring survive to a reproductive age has a higher fitness, and its genes will spread more. This is what's meant by "survival of the fittest".
The fitness benefit of mutations and the changes they cause is often specific to a certain environment: a thick fur will help an animal survive in a cold climate, but will actually be harmful for animals living in a hot climate. An environment's effect on a population's gene frequencies is called natural selection in analogy to animal breeding: animal breeders select animals with certain traits to have more offspring, and a natural environment selects for traits which are best suited for life in the environment in question.
Even if an organism doesn't reproduce itself, it might help the offspring of its relatives to survive better. Since the offspring of related individuals are likely to bear some of the same genes as the helper, individuals which help closely related individuals to survive are likely to further the spread of their own genes in the population, as well. This is called inclusive fitness, and it has the strongest effect on a population when the cost of helping a relative is small and the benefit to the relative is large.
Natural selection does not always work, however. Even if an organism had genes giving it the best fitness in the population, it might still get killed or fail to reproduce due to plain bad luck. Alternatively, in most sexually reproducing organisms, each parent contributes one half of the offspring's genes. Since the contributed genes are random, genes which are possessed by only one parent will not always get passed on, regardless of their fitness. The effect of chance on gene frequency is called genetic drift, and it has a stronger effect if the size of the population is small or if fewer individuals possess the gene. If a group of thousand individuals all have a particular mutation, they're less likely to all die of bad luck than a group of twenty.
If a population somehow becomes separated into two or more groups, any mutations occurring in one group will not spread in the other. If enough time passes and the environments the groups live in are different enough, they'll eventually become so different that individuals from one group can't reproduce with individuals from the other. This process is called speciation, as it splits a population of same-species organisms into two or more different species.
It is important to remember that evolution works neither to directly benefit individuals nor to benefit the species - it is simply a process by which certain genes spread. It can even harm a species, if the species becomes adapted to too narrow an environment which subsequently disappears.