"Graft and corruption" refer to unethical practices that lead to the advancement of those who are willing to cheat to get by. These two ideas have been a part of government almost as long as power structures have existed, as those who are in power use that power to get additional financial and other advantages. In particular, it was graft and corruption that brought the utopian experiment known as the Soviet Union to a crashing halt.
In the movie "Back to School," successful business owner Thornton Melon returns to college to earn a degree. He is in a business class one morning when the professor asks about the necessary steps to open a plant. After the traditionally taught steps have been given, the professor asks if anyone has other steps. Melon raises his hand and talks about all of the functionaries in local government who require bribes in order to make the permit approval process go more smoothly. Because the plant is going to require concrete, Melon indicates that the teamsters union also is likely to need some considerations. When the professor angrily retorts that his class is not about "kickbacks," Melon's response is that these costs are part of real world business. This is just one example of how graft and corruption play a role in modern society.