Corn, one of the last surviving food sources
The Blight is a plague that has ravaged almost all the remaining food sources on Earth. By the time Interstellar occurs, the last crops of okra are dying off, leaving just corn as the only viable food source for humanity.
Considering the blight is a mere plot device, it could be seen as a parable of the hardships faced by humanity in its History and/or as an allusion to the life Great Depression starting in 1929 and the Great Recession started in 2008. With this in mind, the interstellar voyage is an allegory of the hard work needed to overcome great challenges. Difficult times that we overcome is an exhortation to do likewise without losing hope even if all seems lost.
In both versions of the script, a plant disease has ravaged the many sources of food, making wheat, potatoes, and likely soybeans, and rice impossible to grow. With common feed stocks no longer available, eating meat likely became a costly luxury around the world. Corn remains the only viable crop, resistant to the blight, that can be grown and harvested.
2008 script
In the 2008 script, the final result of the blight is depicted. Earth is completely devoid of plant life, with little to no oxygen available. Vast, barren plains, bare mountains, abandoned cities and towns, and lifeless oceans. With no trees to slow down the wind, dust storms blow fiercely, and are strong enough to demolish abandoned houses.
2014 film
Professor Brand explains that wheat became extinct seven years before Cooper's arrival at NASA, and okra was due for extinction the present year. Like the potatoes which led to the Irish Potato famine and the wheat in the Dust Bowl (during the Great Depression), corn was expected to die off as well. Brand explained that 80% of the Earth's atmosphere is nitrogen which humans don't breathe, yet the Blight is known to consume nitrogen for sustenance. Vegetation creates breathable oxygen and as the Blight consumes more plant life and all available soil nitrogen, less breathable oxygen will be available for humans and animals, while Earth's nitrogen will be plentiful enough for the Blight to thrive, allowing it to spread farther and consume greater quantities of plant life. A reduction of oxygen in the atmosphere coupled with the extinction of plant life would make the Earth uninhabitable.
Speculation
Blight could either be bacteria or fungi, however in both versions of the film, the origin and the exact nature of the blight is undetermined, although it could be speculated that it arose from climate change, or the widespread use of mono-culture farming practiced today, or simply a natural, genetic mutation that gave birth an aggressive strain.
Or a disease of extraterrestrial origin, carried by a comet or a meteor.
It is also likely the Blight has caused the devastation of other plant life on earth, not simply edible crops, but grasslands and forests that held down the top soil. Therefore, dust storms could have been the result of the progressive deforestation of planet Earth. In any case, the true origin of Blight is never disclosed.
There is also the possibility of GMOs being a cause of the Blight, on the same basis as Raoultella planticola, which was used to create alcohol naturally, but instead would starve and destroy host plants and could kill many or all terrestrial plant species.