I would like to thank my opponent for starting this debate and look forward to debating someone from my own area.
1) I would like to address the mentioning of Chernobyl, even though it is out of order, because it is the most misunderstood argument that is used against nuclear power. "The Chernobyl accident in 1986 was the result of a flawed reactor design that was operated with inadequately trained personnel."  We also need to note that Chernobyl was a steam explosion, not a nuclear explosion. And was caused by a combination of very poor decisions, from the lacking of basic safety, even for that era, to poor training, to management personnel that was more interested in not looking bad then real issues. The safety issues that happened at Chernobyl were already impossible in the US because of the mechanical safety features we had back in the 80's and our safety has only gotten better.
2) "There is no way to permanently and securely store nuclear waste." I have to disagree with this on two points, first, we don't have to "store" it. The most common nuclear material is U-235 (Uranium) and it is found naturally across the planet.  If we have to be completely safe, it is possible to de-rich uranium (though it is more costly then storing it is depleted and takes up much more space) so that it can be put back in the ground just as it came out, as safe to nature as it has always been. But even without going through that much effort, the storage of fuel is actually extremely safe. The Science behind the yucca mountain shows that it is a safe storage , and actually the main fear is that water will corrode the containers if it rains too heavily over enough decades. However, it is possible to upkeep the containers (since we upkeep all our roads and buildings and everything else from corrosion), and we can even design the containers to be extremely immune to corrosion  so that the life span goes from decades to centuries to possible millennia.
Nuclear power plants also don't produce that much waste, about 20 tons a year per plant.  That's the same amount of waste (by weight, not volume) as about 25 people, per year.  So when we stop and think about it, it really isn't that much.
Going back to safety. Nuclear waste containers have some of the best safety features of any engineered structure.  And to show the kind of tests these have to go through.  They test those buggers for everything! We have PBRs (Pebble Bed Reactors) that add a whole new level on safety.  A PBR is "a reactor whose safety is a matter of physics, not operator skill or reinforced concrete." We also have Fusion reactors which are starting to pick up steam (couldn't resist the old energy pun from the steam power days). And they too have their safety built in the laws of physics  in that if something happens that jeopardizes the power plant, the nuclear reaction will come to a stop on its own, rather then snowball out of control.
That is a simple look at the science of the safety, now lets look at the history of the safety, since this is the only aspect where nuclear power have blemishes, since certain nations choose to let their safety slide, they have become bad marks on the entire nuclear concept. But in the USA, not a single human being has EVER died from an incident related to a nuclear power plant or its waste.  The most famous incident in US history is the TMI incident, which actually should have been a beacon of hope and success. A most basic back story, in 1979, the TMI nuclear power plant had to be evacuated because a partial meltdown began to happen. The wonderful part about what really happened was that the plant managed to cool itself down on it's own. The built in mechanical safety features protected it from anything really bad from happening. While just over the vent stack, readings up to 1,200 millirems were recorded (indicating that radioactive material was making it out into the environment. The government maintained a registry for 18 years of the 30,000 people that lived in the vicinity and found that they did not experience ANY health fallout (no increased rate of cancer, no increased rate of birth defects, no increased rate of brain issues). 
Research has shown that nuclear workers will, on average, be subjected to about 1,000 millirem per year (compared to the average individual who gets about 360 per year), and that will, on average, lessen the workers life by about 50 days, compared to being 15 pounds overweight (lessens your life by about 2 years) or smoking a pack a day (6 years).
It is also only fair to compare this energy source to other ones, such as coal or oil. Those industries can only wish for the environmental and safety history of US nuclear power plants. If my opponent wishes to challenge this argument, I'll provide sources then, but right now, it is getting late, so I'll end this round here.
"Nuclear waste doesn't just go away." I will address this in several ways.
A) The nice thing is, that it actually does, just really slowly. But I see that my opponent did not mention anything about how all this radioactive material is found in the ground [my #2 link, last round and my opponent's #3 link this round], it is a part of nature. It physically can be diluted back to what nature had it as and put back in nature, zero sum.
B) The sunshine canyon landfill can hold 90 million tons of waste. And a new land fill outside of LA will be taking 20,000 tons a day for the next hundred years (700 million tons of waste). First thing to point out is that I'm not advocating these particular locations, only pointing out that we have the space. Since the average nuclear site produces 20 tons of waste a year, 10,000 nuclear reactors (which is over 20 times our current world amount of 436) would produce 200,000 tons a year, and would take 3,500 years to fill that one single location (and we could have more then one, easily).
C) Looking at the potential a 700 million ton land fill. We can also see that if the land fill was full of 700 million tons of nuclear waste, breaking down with a 1/2 life of 324,000 years, means that 350 million tons would break down after 324,000 years, or at a rate of 750 tons per year (1 - (.5)^(1/324000)), or enough for almost 36 nuclear reactors running full time with the amount of waste they produce actually breaking down, so no net volume is added. And remember that is for a single land fill site (we can have more then one and also remember that I am not suggesting that particular site but only the concept that it is physically possible).
This doesn't even go into detail the possibilities of the future. Ideally, waste disposal wouldn't be done on Earth. Waste would be ejected from Earth, into either Jupiter or the Sun. Many look at that as too futuristic to be plausible, and in some ways it is true. We do not have the technology to make those idea economically or logistically feasible, yet. When we stop and look at how far aerial travel has come in the last 100 years, or how far space travel has come in the last 50 years, we know that where we will be 50 years from now is going to be unimaginably faster, safer, and cheaper then it is right now. Now 50 years from now is something that we can easily do for nuclear power, heck we have the technology that we can wait for hundreds of years before running into trouble, even thousands.
But one can stop and ask "why? why do we need this?" that is simple. For one, it is cheaper for the people to be powered by nuclear power then coal (our current main power source) and so will save people and businesses money and that will lead to a better economy. That right there is more then enough justification to answer the "why," but the other thing that allows is to free up millions of tons of coal that is currently being devoured by our power plants. This coal is then free to sell to other nations that still use coal and so bring in billions of dollars in exports (benefitting our economy even more). And the selling of coal to third world nations will allow them to power their nations and so move our of the dark ages. Once out of the dark ages, they can begin to go through the industrial revolution and have jobs for their people and that will allow their economy to blossom. Once that happens, we then have more nations that we can sell our other products too, making for even more exports for us and more help to our economy. So in the long run, our economy benefits 3x (the nuclear that saves money, selling coal to make money, and the emerging markets to make money) and other nations benefit by going through the industrial revolution.