I'm actually applying for Nuclear Engineering as a course, so this is an interesting thread for me.
1. I feel the public has been misled, but it is a tough subject that requires external reading to fully appreciate. There are lots of concerns, whether waste or proliferation or safety. The problem is, all three of these concerns revolve around the design of the reactor. The costs of nuclear seem artificially high, due to regulatory requirements.
In any nuclear discussion, Nuclear Energy comes in two main flavours - fission when you split up large atoms, or fusion when you fuse small ones as the entire Universe wants to make iron. This post is about Nuclear Fission.
I'll first define a few terms.
Radiation simply refers to fast-moving particles, and the types coming from radionuclides are called ionising radiation. If it hits a cell, by analogy it is like opening up your DNA file in notepad and editing the code. It happens all the time, the sun emits ionising UV radiation and the granite beneath you produces some radon gas which is radioactive. It's okay, except once in a while a cell becomes too damaged or cancerous.
Penetrating power refers to how well it could penetrate through various materials, range refers to the distance travelled before the energy of the radiation is near zero, and the ionising ability refers to the chance of ionising something when it passes through.
Alpha particles can be stopped by a thin layer of skin (helium nuclei are relatively huge) - there's no harm in holding an alpha source, but if you ingested it then it would immediately ionise the nearest thing (meaning it damages your vital organs). It is important to know this because Fukushima Daichii released alpha-emitting iodine into the atmosphere.
Gamma radiation has infinite range (it's a high energy form of light), high penetrating power (requiring thick lead or concrete to block) but is unlikely to ionise anything when it passes through you. It is still quite dangerous when exceeding certain tolerances though.
Beta radiation (fast electrons) is moderate on everything.
Nuclear waste is a complicated issue, but you have to understand that not all waste is equal in quite a few ways. Suppose you had two pieces of nuclear waste material, one which lasts 10 days and another a million years, which one would you presume is more likely to hurt you? If you guessed a million years (intuitive), it would be deadly wrong.
I'll ask another way. If I told you that a candle and a dynamite releases about the same amount of energy, which one do you think would hurt you if you held one when lit? It's obviously the dynamite, because it explodes - the energy rushes out all at once, and the energy doesn't get dissipated. It's the same principle with radioactive materials, short half lives usually mean a greater hazard
The stuff that lasts a million years, you could probably make a plate and eat off.
The other issue is the quantity of waste. If an average person with a US/European standard of living needs about 30kWh of electricity, what does it take to produce it? A majority of power is produced using finely ground coal-powder. It takes about 30 pounds of coal to produce your electricity needs, which is about a large rucksack full.
The combustion of which emits NOx, SOx, CO2, and other harmful pollutants like heavy metals and soot into the atmosphere. It also emits radioactive "NORMs" (naturally occuring radioactive materials). It does get diluted, but very large numbers in the hundreds of thousands have diseases (cardiovascular and respiratory) directly related to air pollution mainly from power generation and motor vehicles.
Current generation power stations (III or lower) need only hundredths of a gram of U235, or about 4 grams of natural uranium for the equivalent amount of energy. The reason why the numbers are so high is because over 95% of the fuel is being dumped as "waste" - it's actually not waste at all.
A lot of the waste issues are born from the fact that the fuel is put through once and disposed of, if it was properly reprocessed or "bred" then we could use up all the energy in natural uranium instead of less than 1% of it (the percentage of U235, the fissile isotope, present). It is politically controversial because there may be proliferation risks associated depending on the exact details and designs of the reactor, which is usually too technical for Parliament to interpret.
A coolant refers to the medium that carries the energy from the nuclear material to the thermal exchange. The safety issues are mainly to do with the water used as a coolant. If you heat up a pot of water, it gets hotter and hotter and when it reaches 100C, it starts to boil. The steam takes up a lot of volume, about a thousand times that of water.
"Gen III" or lower reactors, referring to the current generation of Nuclear Power Plants, are almost exclusively based on water-coolants. PWR is a "Pressurised Water Reactor", BWR is a "Boiling Water Reactor", and they are both subclasses of LWRs - "Light Water Reactors". Light Water contrasts with Heavy Water which contains hydrogen atoms with a neutron attached to it, called deuterium.
Heavy Water slows down neutrons by colliding with them, which increases the chance that the neutrons will hit a radioactive nucleus - it acts as a moderator.
The entire design of a nuclear power plant is based around this issue - it has to have something called a containment building which is a large concrete building surrounding the reactor/pressure-vessel should the reactor fail and a breach occurs (usually when pressures are too high). It is what contributes to the cost of construction, decomissioning, and environmental/safety issues.
The design of current nuclear reactors require an active power source to ensure there is no meltdown, which seems quite silly after Fukishima Daiichi.
In Fukushima Daiichi, a tidal wave breached the sea wall (which was constructed too small as it was deemed an unlikely event during the lifetime of the reactor) and caused what is known as a common mode failure where multiple backup systems failed simultaneously. The failures led to its eventual meltdown and several hydrogen explosions caused by the fission of water and a heat source in the reactor.
Proliferation deals with the part that involves enrichment, as 1% uranium won't burn the same way coal diluted with sand doesn't like to burn. If it's possible to enrich to 5%, then it's also possible to enrich to above 90% (weapons grade). This issue is notoriously difficult to manage.
A lot of these problems are inherent to a particular design, and as it takes many years to research/develop alternative fuels and coolants and designs and further to approve/construct/connect-to-national-grid/export - nuclear energy won't be popular until it becomes clear that it is practically impossible to reach the Paris/Kyoto climate targets without it, the cost of energy to rise, and the intermittency of renewables cause the lights to go out more often.