The reason why nuclear waste is bad is that it has the potential to be catastrophic for the environment. With nuclear power plants, there's always a chance for human error, so if a reaction were to get out of control it could contaminate our environment. Thankfully, U.S. nuclear reactors are operated under many strict conditions that reduce this risk.
The most prominent danger presented by nuclear energy is the radioactive waste it creates, like uranium mill tailings and spent reactor fuel. These retain their radioactivity for thousands of years and remain a threat to life on Earth.