The waste that contain radioactive material are normally called as radioactive waste. The remains that are available from nuclear energy power plant are called nuclear waste.
The waste, sometimes is dangerously radioactive, and remains so for thousands of years. When it first comes out of the reactor, it is so toxic that if you stood within a few meters of it while it was unshielded, you would receive a lethal radioactive dose within a few seconds and would die of acute radiation sickness within a few days. In practice, nuclear waste is never unshielded. It is kept underwater (water is an excellent shield) for a few years until the radiation decays to levels that can be shielded by concrete in large storage casks.