This old youtube of Kirk's lecture has been making the rounds. Most of this post will regurgitate his major points for those without three hours to kill.
First up, the two major reasons our nuclear plant designs suck so bad, followed by miscellaneous stuff:
Water Cooling
Okay. Water. As a coolant. Think about that for a minute: what temperature does water turn into a gas and stops being water? Like 212 F, right? And how hot does a power plant have to be to efficiently turn turbines?
For water to stay as water, it requires a system in place to put it under pressure to increase its boiling point. This use of high pressure and water is the #1 safety threat in these plants, and is a huge chunk of the cost of a plant. The large concrete sarcophagus of the containment building has to be several times larger than the reactor to contain the expansion of water into steam in a worst case scenario.
Fukushima demonstrated best how dangerous water is - when sociopathic idiots allow the pressure system to fail [1], the water turns into steam. The steam separates into hydrogen and oxygen. The hydrogen and oxygen separate into separate layers, with hydrogen at the top, and the hydrogen eventually decides to explode if it feels like it's too hot.
Salt as a coolant has been knocked around as an idea worth using. In a salt cooled reactor, in the worst case scenario where God comes down from the heavens, rips the building and reactor in half: The fuel and coolant will spill out in the room a few feet in all directions, then cools into a solid on the floor; never turning into a gas to spread all over the place and fuck up everyone's potatoes. In a more reasonable scenario, the reactor suffers a puncture or tear that seals itself shut, and the contents of the reactor are emptied into an underground emergency storage tank that's designed to shed heat, turning everything into an inert solid.
Solid Fuel
Our reactors use solid fuel. Does that make sense? That does not make sense.
From a safety perspective, this doesn't make things too much worse. You just have a missile shield in place to make sure that the inevitable hydrogen explosion caused by a literal real-life Mr.Burns, doesn't launch them out.
There are a lot of down sides to this however:
* You can only burn around 0.5% of the fuel before pulling it out, since if it shatters into fragments you're screwed. The Uranium needed for fission isn't an infinite resource, and you're talking about making the stuff at least a couple orders of magnitude more expensive to use than it really is.
* Additionally it's kind of hard to have a "melt down" if you're using liquid fuel. Being as how it's already "melted" to begin with..
* You can't separate out the elements generated by all that fissionin'. This stuff isn't worthless green goo like in The Simpsons; they're just elements. With extra neutrons in them. And decay chains to get those neutrons down to stable levels. They're rare stuff in nature, and there's lots of applications for them; medical, fuel for deep space NASA probes, etc.
* Fission has a nemesis: Xenon. Gobbles up all your neutrons and doesn't give anything back. In liquid fuel, it's an impotent gas that will bubble out and float far far away from all your precious exploding atoms.
* It's a bit of a razor blades model. Hey buy our power plant, oh yeah, and you're going to have to buy our fuel for the next 50 years.
Breeder/Burner
A burner reactor is simply a reactor that burns fuel as you put into it. Really not much different fundamentally than a furnace; just you're using fission instead of combustion to generate heat.
A breeder reactor creates fission material as it goes. With Thorium, you split uranium, this releases some neutrons, the thorium absorbs neutrons, and then transmutes [2] into uranium from beta decay, which is when a neutron turns into a proton.
This is ultimately pretty useful, since there's a finite amount of Uranium 233 out there. Getting more out than what you put in can be used to kickstart more reactors.
Alternatives
Alternatives to decent nuclear reactors are... somewhat lacking, on a number of fronts. Foremost in the media and public consciousness is nuclear hysteria. Nothing in the universe is free. Coal plants for example, spill out tons more radiation than nuclear ones, and kill far far more people all the way from the supply chain to exhaust. As the always sexy Matt Bors puts it:

But we've got to be brutally honest here; human lives don't matter all that much. If they did, we'd outlaw cars and would ride bicycles to work instead. What really matters is getting access to that sweet, sweet energy. Peak worldwide supply of oil and coal are coming up, we've had several wars over access to this stuff in the past century, global warming can cause us to all starve to death; put simply, the status quo on power isn't going to last this century.
Renewables have a lot of problems. Foremost is energy density. Sunlight isn't even fierce enough in atmosphere to melt snow on a road. This means you have to spend tons of money making gigantic power plants - you basically need the space of a very large city to get from solar what you could get out of a thorium breeder reactor in one warehouse sized spot. [3] This also causes cost issues in additional transmission lines to transport the electricity.
Exposure to elements is another. Tree leaves are death to solar panels, so you need to kill anything alive within a mile. You've got to have guys wipe off dust from a field of panels as large as New York City. The power generators will just die much sooner if they were in a safe cozy concrete bunker.
And storage is extremely unrealistic currently. You're going to spend a dollar to build a battery to store $0.00001 worth of electricity? That's.. not very efficient.
For base, cost effective electricity generation that can be used anywhere, we have to spin turbines with some kind of heating element. Wind and solar can only be a supplemental supply [4]. We're stuck with that.
Energy Means Prosperity
One of those things batshit people (example: Helen Caldicott) suggest we should do is... basically tear down civilization altogether. Live like the Amish. Things will be better this way.
I'm of the opinion... things would not be better this way. Disregard about how this means we'll never get those robot women or space colonies or jetpacks science has always promised us... and note how much better life is with mechanical power. The difference between the health of a rich first world country and a starving third world [5] one isn't some special snowflake difference in the people living there, but access to fuel.
Note the foundation of every economy: food.

Crops such as corn and wheat are now largely done by one guy driving over a field a few times. They're so cheap, governments have to pay people to grow them. [6]
I'm not saying the place is going to turn into a Mad Max style hell hole (as opposed to an Amish Paradise), but it's pretty realistic to expect we'll eventually revert to the biblical style tribalism that we still see in places like Afghanistan. [7] That means a good chance that slavery and ownership of women will make a comeback. You can choose to believe that the 1950's prove what magical extra special people we are compared to most of the history of humanity, or you can believe that times of prosperity is the only way people can have the luxury to be decent.
The Library of Alexandria has already been burned to the ground a couple times over; I don't think it's a good idea to do it again.
Future Developments
If it's such a good idea, why haven't/aren't we doing this?
Well, basically because only governments can make it happen. If you own $trillions of oil in the ground, or you're already running a nuclear plant, there's no incentive to spend $2 billion and a decade on R&D to render the assets you already own less valuable. In the US, Weinberg was fired by the Nixon Administration [8] for squawking incessantly about science, the laws of physics, future of humanity blah blahbitty type stuff.
So uh. Eh. There are some companies that are trying to get to liquid fuel, but uh..... yeah guess what.
China is re-developing the Thorium breeder reactor. And plans to begin deploying it 10 years from now.
So uh, in the future the heart of all of our power plants will be bought from China? [9] Let's pretend to be surprised if that happens.
[1] Seriously, did even a single person who made the decision to keep the emergency generators in a nice low area to get knocked out by a tidal wave ever see prison time? Or was BP's strategy in these matters sufficient?
[2] You may remember that transmutation was that thing alchemists wanted to do to transform lead into gold. As far as we know, the only viable way to do this is with nuclear fission or fusion. In the long long run, this sort of elemental forging could be pretty important. Scarcity is one thing, but another is that every element from tasty Bismuth up is unstable (ie, "radioactive") and not long for this universe. We don't even know if hydrogen is going to survive.
[3] The density of fission versus combustion is ridiculous - a city like San Antonio would have its electricity needs met by one or two truckloads of thorium. Per year. That's a lot of lost jobs in transportation.. or several orders of magnitude more power.
[4] This is still very valuable. Much like water, electricity has marginal returns. Some electricity is much, much better than no electricity.
[5] We've really no right to say "hey, no western style of living for you." I'd go on record as saying that's intrinsically immoral.
[6] For crops that need to be harvested by hand, we pay immigrants around 3 to 7 dollars an hour for the type of job that's comparable to mining coal, as far as its health effects go.
[7] A place where many practice biblical law.
[8] An era where they also thought space ships were too awesome and cheap so they decided to scrap them all and make crummy shuttles. All so Putin could make trampoline jokes at our expense decades later.
[9] Versus just building more coal plants. Which is... really optimistic of me to even suggest, knowing how my country works. Eh..