Nuclear power is made by combining (fusion) or breaking apart (fission) the nuclei of atoms. There are many ways to do this, with many different kinds of atoms. The material that you select has huge implications for the process - its outputs and its dangers.
In the 1940's, scientists developed a fusion process for hydrogen, which causes a chain reaction that's nearly impossible to control. And the world's first view of nuclear energy was one of destruction, suffering and out of control human carnage.
Over the last 50 years, there have been two major accidents, which further damaged the public's opinion of nuclear power. At Three Mile Island, the reactor overheated, causing radiation leakage. There were safety features in place which prevented the accident from being much worse than it was. With the advances made since then, containment buildings can be made even better.
The worst accident by far was Chernobyl in 1986. The containment structure was poorly designed and thousands of people eventually died from radiation. Again, this is quite solvable.
Nuclear power plants today use a fission process with uranium. They can control the process by how much raw material is added. If they stop adding uranium, the process slows and stops. Nuclear power is being used cleanly and safely by many countries, including the U.S.
Dangers include mining, transporting and storing raw materials and storing waste products until they're no longer dangerous. These risks can be managed. Another risk that concerns people is that terrorists could wreak havoc if they got some of this material. While this is true, they can also cause a great deal of trouble with an airplane. It's easier to lock up nuclear materials.
Nuclear power can be created with a variety of different materials (remember the periodic table of elements?). Many are readily available and lend themselves to a controllable process. Nuclear power is very clean in terms of carbon dioxide emission. There are risks to be considered, but they can be controlled, as has been proven over the last fifty years.
We need to look at alternative power sources, including nuclear power, in a rational way. The fact that people's first exposure to it was a spectacularly destructive bomb is unfortunate. There are many things in this world that are very useful but can be terribly harmful under the wrong circumstances. (Ever heard of di-hydrogen monoxide?)