Technically, corruption covers an entire host of abuses, of which graft is one. Graft and corruption are charges that are typically leveled at highly-placed government officials, who are able to use public funds to improve their own fortunes. When an official uses the access, influence, knowledge or power that comes with an elevated position and engage in graft, they abuse public trust and may be charged with a crime.
TL;DR (Too Long; Didn't Read)
Corruption is a broad set of behaviors based around the dishonesty or unethical conduct of a person in authority. Graft is a specific type of political corruption, where a politician uses her authority for personal gain.
Corruption Covers Many Types of Dishonesty
The vice of corruption includes many possible acts where the honesty, integrity or moral principle of the perpetrator is compromised in some way. The word typically is used to describe the unscrupulous acts of politicians and government officials. Broadly, corruption is a departure from the original or pure form of a thing or idea and includes such acts as bribery and embezzlement.
Graft is Grease for the Machinery
Graft is a specific type of corruption whereby an official uses his public stature to gain illegal benefit. For instance, a senator who sits on the armed services committee in the U.S. senate cannot use his knowledge of military contracts to buy stock in a defense contractor's company. His position gives him unfair advantage over other investors. It is similar to the notion of insider trading in business. A strong example is the trial of former Illinois governor Rod Blagojevich, who was accused of soliciting cash from suitors in exchange for appointment to an empty senate seat that he controlled.
It Can be Hard to Tell the Difference
It can be difficult to tell the difference between graft and other types of corruption. In some cases, the differences are slight. For instance, the main difference between bribery and graft is the element of public finance. A police officer who looks the other way because criminals pay him off is accepting a bribe. A senator who approves a military budget that includes $300 hammers and $600 toilet seats is engaging in graft. Penalties depend on the type of corruption that has occurred but a typical penalty for willful involvement in graft would be five years' imprisonment.
Role of the Watchdog
Exposing corruption can be difficult. Appointing internal agencies to keep an eye on corruption may simply lead to more opportunities for graft. One reason corruption is more difficult to hide in democratized countries is freedom of the press. In societies where the press is under government influence, investigations of corruption can be easily quashed. Often, citizen-led groups are formed to keep an eye on specific industries or government processes. These are called watchdog groups, and they rely on independent, generally non-profit charters to preserve a reputation of integrity.