The human population started booming in the 19th century with the improvement of common hygiene and medicine techniques. In the 20th century, the rate of population growth was exponential, meaning that it was frequently doubling. With the increasing consumption of resources and expansion into new territories, a bigger population inevitably means a bigger impact on the environment.
Many problems like extinction, depletion of drinkable water and general pollution are related to this problem. At a certain point, the planet will no longer be big enough to absorb all the impacts.
Overpopulation doesn’t only apply to humans; it also concerns other animals, which are part of a food chain. When the population of a certain species becomes too high, the one under it on the food chain (their food) becomes rarer. It works the same way for humans, but since they consume all type of resources, the impact is practically on the entire planet. Not only that, because humans have no natural predators, there is nothing to balance the numbers.
Painting by Dave Todaro, http://nuvango.com/davetodaro
How do you manage overpopulation? We have to look further then just using condoms and contraception pills. Some countries like China and India are able to establish laws to contain the population growth. They call it family planning and it gives an advantage to a family having only one child.
However, what we observe in the world is that countries that are well developed tend to have a stable or declining population. Education to both sexes (especially women), birth control and increasing wealth would be three elements that help containing the population growth. Until those elements are spread on a larger scale, the population growth in the world will keep accelerating, resources will become rarer and pollution will keep increasing.