Graft, as understood in American English, is a form of political corruption, being the unscrupulous use of a politician's authority for personal gain. The term has its origins in the medical procedure whereby tissue is removed from one location and attached to another for which it was not originally intended. Similarly, political graft occurs when funds intended for public projects are intentionally misdirected in order to maximize the benefits to private interests.
Political graft functions when the public coffer is directed to purchase a good from a specific private interest at a cost far above regular market rates. The private interest then siphons some of the gratuitous profits to government officials that are able to ensure that future government spending continues in the same fashion so that this lucrative relationship continues. A member of a government may misappropriate directly from government funds, but they may also make decisions benefiting their own private economic interests by using inside knowledge of upcoming government decisions to their benefit, in a manner similar to insider trading.
Although the conflict between public and private interests is common to all forms of corruption, the term "graft" is specific to the intentional misdirection of official funds. Although not the original usage of the term, graft in the modern context is commonly, but mistakenly, used as a blanket term for political embezzlement, influence peddling or other forms of corruption. While embezzlement and influence peddling are elements sometimes present in graft, the relationship is not deterministic.
Graft is exemplified in the film Mr. Smith Goes to Washington: A local political boss has insider knowledge of a government dam building project that will necessitate the purchase of certain tracts of land. In order to make a hefty profit, he purchases the land in advance so that he may charge the government exorbitant prices to buy the land it needs to complete the dam.