Nuclear energy is an important contribution to our country's electricity supply, making up about 20% of all electricity production. It offers what is called "base load" power, which means it provides large amounts of electricity all of the time, but isn't designed to turn on or off very quickly.

Nuclear power production is attractive because it does not release any greenhouse gas emissions like carbon dioxide or methane. Like any electricity source, though, there are environmental costs to nuclear power, including uranium mining, water usage for cooling the reactors, and production of radioactive waste.

So, to answer your question directly, we don't need nuclear power, and many countries don't use it. If we chose to stop using nuclear, however, we would either need to replace all of the power that those nuclear plants produce with other sources of electricity, such as coal, natural gas, solar, etc. or dramatically reduce the amount of electricity we consume.

Depends on what you mean.
Nearly all of the energy that we humans use is nuclear in origin: the sun, which provides the vast majority of our power, is a nuclear furnace, basically a giant self-sustaining hydrogen bomb. Most of the rest of our power comes from the geothermal energy inside the Earth, which is driven by radioactive decay, also a nuclear process. The only source of energy that I can think of that isn't either directly or indirectly nuclear is the tides, which are gravitational (fossil fuels are indirectly nuclear because the energy that made the fuels in the first place came from the sun).

Nuclear power generated in nuclear power plants is a lot more iffy. It does not produce greenhouse gasses, which is good, but it does produce radioactive waste. There are ways to dispose of radioactive waste that as far as we know are clean and safe if all goes as planned, but whether all will go as planned is a question of debate (mostly political, but there are scientific arguments as well).