A line-item veto is the power of the president (or any leader of an executive branch) to reject (veto) certain individual parts ("line items") of a piece of legislation (a bill) without rejecting the whole thing.
In the United States, almost all governors (leaders of the U.S. states) are able to use the line item veto. Currently, the President of the United States is not able to use the line-item veto. President Bill Clinton was given the line-item veto by the United States Congress for a few years until the Supreme Court said that it was unconstitutional
People who like the line-item veto say that it is good because it allows the President to remove unimportant waste from important legislation. For example, it would be too dangerous to completely veto big pieces of legislation (like the budget for the military or the budget for other government departments), but with the line-item veto, the President could choose to keep what he or she thinks are the good parts of the bill and reject the bad parts of the bill.
People who do not like the line-item veto say that it is bad because it gives the President too much power over Congress and believe that it goes against the checks and balances created by the U.S. Constitution. Other criticisms include the possibility that the president's individual vetoes of line items make the bill different than what Congress voted for.