Biological evolution can take place in many ways. Many of the other writeups here describe the process in terms of passing along those traits that allow an organism to survive for longer. In theory, organisms that avoid being eaten will have more opportunities to reproduce. That is how we end up with insects that look just like leaves and zebras whose markings make it difficult to tell them apart when they stampede.
As quincyfree described earlier, a species can also evolve as a result of genetic drift and mutation. A salient example of this is the lack of genetic variability between harbor seals. This species was hunted almost to extinction; recently, the number of harbor seals has increased a great deal, but because the current population was derived from a very small population, the genetic difference between individual harbor seals is very small. Genetic drift and mutation are the main causes of evolution in species that reproduce asexually.
One additional force that drives species evolution (at least of sexually reproducing organisms) is sexual selection. Sexual selection works independently of what has so far been referred to as "natural selection," and favors those organisms that have traits that allow them to reproduce more often. Basically, sexual selection acts to preserve the heritable traits that make a particular organism sexy. The classic example of this is the peacock's tail. If you have ever been around a peacock, you know that the big fancy tail is little more than a convenient handle. However, peahens find that same elaborate peacock tail practically irresistable. It is possible that sexual selection is responsible for the entire population of people that are cute, but stupid.
So, in the end, there are basically three traits any individual organism can have that increase the chances that its genes will make it into the next generation. It can be smart (natural selection), it can be lucky (mutation, genetic drift), or it can be attractive (sexual selection).
Good luck transmitting those genes!