Veto power is a very important power granted to the executive branch of government. This would be the president at the federal level and the governor at the state level. The word "veto" means to reject something. Having veto power means the president or a governor can reject a law passed by Congress or the state legislature. This could prevent the law from going into effect.
This power is very important because it forces the members of the legislative branch to consider the thoughts of the leader of the executive branch. If the president hints at vetoing a law, Congress may try to work with the president to see if there is a way to write the bill that the president can support. The same would be true with the governor at the state level. Overriding a veto is a very difficult thing to do.
Veto power is part of our system of checks and balances. It helps to keep one branch of government from becoming too powerful.