In the strictest sense, a corrupt politician is someone with political power who uses that power immorally or illegally in order to attain some kind of personal benefit. The benefit could include anything from money to additional power. According to many definitions, political corruption has to involve some kind of financial element at the core of the behavior, and this is primarily what separates it from the general term "abuse of power," which is often more broadly defined. In terms of actual behavior, some of the things that would be typical of a corrupt politician include accepting bribes from private donors, offering jobs in powerful positions in exchange for favors, and manipulating legislation inappropriately in order to benefit the politician's own private business interests. There are many problems associated with corruption in politics, mostly related to the fact that a corrupt politician will often act strictly according to his own interest, ignoring the interests of his constituents and the society as a whole.
Corruption in high office is a classic human problem that goes all the way back to ancient times. In modern political systems, there are often many protections to reduce the amount of political corruption with laws and restrictions set up specifically as safeguards. For example, term limits are sometimes recommended so that politicians don't become overly concerned with maintaining power for a lengthy period, and some countries have laws about private donations to political campaigns to avoid overt bribery.
One of the classic activities often associated with a typical corrupt politician is the willingness to accept bribes. For example, a private company might secretly transfer money into the bank account of a politician as a reward for voting against a legislative measure. Sometimes this kind of bribery is handled in a very subtle ways so that the transfer of money isn't as obvious or direct. For example, a politician may receive an especially good deal on a mortgage from a bank after agreeing to vote against a banking regulation bill.
Other common activities a corrupt politician might engage in include cronyism and legislative manipulation for private gain. Cronyism is when politicians give people powerful appointments for some kind of favor, often including those who offer bribes and other secret favors. An example of legislating for private gain would be if a politician were to change the law so that he could make more money in his private business by getting rid of regulations that protect the public.