Evolutionary Arms Race (alternatively known as the Red Queen Effect) is a term that refers to species evolving competitively, particularly in reference to interaction between mammals and bacteria/viruses. As the microbes evolve to get around the defenses of its target creature, that creature tends to evolve defenses of a proportional strength. Should either faction develope a mutation that confers a significant advantage, the other faction will tend to be diminished, at least for a period of time. This microbe-macrobe interaction is the subject of an hour long PBS special.
This competition need not be interspecies; it can also be sexual. In most species, the male benefits most by mating with as many females as possible, thus gaining maximum genetic representation in the next generation. The female, however, can only produce a limited amount of offspring at one time, and therefore wants to have a male around to help protect the offspring. With these rather contrary goals, both sexes are constantly developing methods by which they might better attain their goals, but with any given species having so large a population, neither sex is likely to gain a significant advantage. Even if, by some astounding mutation, one of them did happen to gain a significant upper hand, it would tend to unbalance the species, leading either to extinction or to a balance of power between the sexes similar to what had been seen before.
When there is a conscious force, such as humanity, acting in the scenario, things become markedly more complex. It turns out that while we attempt to develope antibiotics and vaccines to fight disease, we are actually just screwing ourselves over; those few strains of the microbes that are resistant to our drugs are the ones that survive and propagate, thus strengthening their effectiveness. Take, for example, the death rates due to pneumonia and influenza over the last half-century. When first began to use antibiotics, those death rates dropped drastically, but began to rise shortly after. They are still rising.
And that is a best-case scenario. An example of a far worse variation can be seen in tuberculosis. For a while, we had tuberculosis "beat", but then it resurfaced in the Russian prison system, only now it is ultra-resistant to our common methods of fighting it. Take into account the fact that tuberculosis can be passed on before the carrier is aware that they have the disease, and you see just what kind of monster we have unleashed on ourselves.
For all of humanity's understanding of nature, and the power that we exert over it, the conclusion that seems to invariably manifest is that, the more we fuck with nature, the more it fucks with us. I therefore decry those scientists who would seek to "win" the evolutionary arms race, for in so doing they weave a noose for ours and others' species.