(See The Effects of Nuclear Weapons, and also "The Pentomic Army" for sources on this)
The 200-300m quoted distance is the "100% probability of kill" range, I believe. Double that range, and the probability halves.
You also have to remember that Hiroshima was almost perfectly designed to be obliterated in a nuclear blast. The topography is that of a bowl, so overpressure actually wraps around rather than just releasing in an outwards pattern. Also, a lot of the buildings were made of very weak materials - residences had a lot of paper and wood, which a) burned really well, and b) did little to absorb the blast overpressure on the way through.
Nagasaki actually fared quite a bit better, as have various test ranges around the world.
In a modern concrete and steel city, the reflective/absorbitive properties of building materials considerably reduce the spread of blast overpressure on a lateral trajectory. Additionally, few cities are built inside a bowl (New Orleans excepted!) - so most of the time, the overpressure only hits you once.
There really are only four lethal mechanisms that accompany a nuclear blast inside the atmosphere: prompt radiation, fireball, blast overpressure (and sometimes a secondary overpressure from air rushing in to fill the resultant vacuum), and residual radiation.
Prompt radiation travels in a straight line, and is blocked quite effectively by earth, heavy metals and some types of clay. At larger distances, even curtains can help with the flash. If you are in direct line of sight to the flash, within lethal range - you are dead. If not, you're probably ok - and the radiation types released in the flash typically don't stick around.
The fireball is typically not very large, but will incinerate whatever it comes into contact with. Most modern designs try to air-burst, and the fireball often won't ever touch the ground.
Blast overpressure hits just like a conventional explosive: a sphere of rapidly moving blast pressure, reducing in power over distance, and also losing energy as it hits things. The same protections against prompt radiation help here: a good wall of dirt does wonders for stopping overpressure, whether it's from regular artillery or a nuclear explosion. Note that studies have shown that blast overpressure is the primary kill mechanism for regular nuclear bombs, just like any other bomb.
Finally, you get residual radiation. This can be avoided almost completely with a carefully designed airburst - most "fallout" and residual radiation occurs when dirt is sucked into the fireball and irradiated there. Burst high enough to not have the fireball encompass a lot of dirt, and you don't have very much long-term radiation. It's largely unknown what the long-term effects of residual radiation are; the area around Chernobyl didn't behave at all like the models we had!
Then there are different bomb designs to consider. A really small nuclear bomb behaves a lot like a really large conventional charge: you could set it off in a football stadium, and probably not worry too much about damage to buildings a few hundred yards away; man-portable nukes were designed on that assumption, as were things like the horribly-design Davey Crockett round.
"Neutron bombs", which really should be called "reduced blast bombs" focus on enhancing prompt-radiation release at the expense of a MUCH smaller blast/fireball (and consequently very little residual radiation). Why would you want to do that? a) It greatly reduces long-term contamination of your target area (meaning you might get to go there!), but more importantly b) it's FAR more effective at taking out tanks and similar. Tanks are really, very, very good at withstanding blast overpressure (it's pretty much their primary defensive purpose - survive artillery and shells while they move forward). It's not at all practical to burst enough regular nuclear weapons to reliably take out a distributed, dug-in tank force. However, they are almost entirely made of metal - and prompt radiation does a "wonderful" job of frying everyone inside them. So neutron bombs give you a chance to wipe out tank divisions, probably with less damage to neighbouring towns (which is why they were invented).
(As an aside - duck and cover is more effective than you might think because of the items stated above; when blast-overpressure is your primary kill vehicle, the same techniques you would use against a regular bomb, or a tornado, are really quite effective)
Lastly, you have "dirty bombs" designed to contaminate an area as much as possible (a pretty stupid idea, unless your objective is revenge), and "enhanced blast" bombs which try to maximize blast-overpressure and minimize other elements - for example, nuclear penetrating rounds designed to blast a bunker as hard as possible.
Weirdly enough, nukes aren't even that effective militarily. In Gulf War 1, a study was commissioned to see how many tactical nuclear weapons it would take to effectively destroy Iraq's dug-in entrenched forces. The answer was in the thousands, simply because they were dispersed and the blast-overpressure kill radius against tanks isn't very good. It was actually significantly cheaper to use conventional precision rounds.
A second study demonstrated that a huge fuel-air explosive is actually both more destructive and more contaminating than a small nuke.
Nuclear weapons aren't nice, and do a lot of damage - but there's a great deal of hysteria surrounding them. Yes, they make a mess. No, they won't end the world (tangent: nuclear winter theory, btw, was debunked and withdrawn by its proponents), or even render a country the size of the USA or Russia uninhabitable/in the stone age.