Nuclear waste affects the environment primarily because it's extremely difficult to dispose of properly. If it isn't disposed of properly, it can cause extensive groundwater and soil contamination. The elements that make up nuclear waste often have long half-lives, which means that it may take millions or billions of years before the waste is safe for humans to be around.
The main issue with nuclear waste is that it's difficult to properly store. There's no proper way to dispose of the waste, so any nuclear waste must be managed until it decays completely. Initial storage of nuclear waste during the Cold War wasn't properly done, so areas surrounding nuclear storage facilities have been found to suffer extensive environmental damage.
The three main types of nuclear waste are low, intermediate and high level. Low-level nuclear waste comes from hospitals, laboratories and industries and is typically not dangerous, although it must be handled more carefully than ordinary garbage. It's normally buried in shallow landfill sites and may be compacted or incinerated. Intermediate-level nuclear waste may require unique shielding and is normally comprised of resins, chemical sludges or reactor components. Depending on the longevity of the intermediate-level waste, it may be buried or disposed of deep underground. High-level waste may refer to nuclear fuel or the waste created from processing it. High-level waste is sealed inside stainless steel canisters and disposed of underground.