Two dangers the Earth faces as a result of overpopulation include the spread of disease causing pandemics and lack of energy sources. Other dangers include possible global warming and Starvation. Overpopulation is a generally undesirable state where an organism's numbers surpass the carrying capacity of its habitat. Read more at http://en.wikipedia.org/wiki/Overpopulation.
Along with the upper answer, there could also be extreme environmental damage to ecosystems and animal habitats. Leading to extinctions. Also there is the high demand for food which can lead to economic changes. Not to mention all the economic changes and conflict between everyone.
Everything the others said is already coming to pass. When any species, including humans, exceeds the carrying capacity of its environment, that species is bound for extinction. That's an immutable law of Mother Nature. We are using resources at an un-replenishable rate and our waste is overwhelming the natural environment.