To supplement niels's answer, the hardest part about a nuclear bomb is to prevent it from blowing itself apart before it has completed fission (or fusion). If a nuclear bomb does detonate, any nearby potential fission/fusion fuel will simply be blown away and not fizz. To understand why we need to understand how nuclear bombs work.
An explosion is a rapid increase in volume and energy causing high pressure, usually due to gas rapidly expanding. A bomb is basically a device which uses extreme heat to vaporize material turning it into a rapidly expanding gas. If this expansion causes a supersonic shock wave we call it a detonation. High explosives are explosives which detonate.
Normal explosives happen via combustion, the technical term for burning. Once sufficient energy is applied the chemicals in the explosive combine with oxygen. This reaction is exothermic meaning it gives off energy. That energy then starts the nearby material reacting with oxygen. Once one bit starts burning it can provide energy to the next bit and the next bit and so on to make a chain reaction. This is why a burning fuse works; so long as there is combustible material, oxygen, and energy to start the reaction, any amount of material will burn.
Nuclear fission works very differently. When fissile material is bombarded with neutrons, some of them will smack into a nucleus and break it apart. This reaction is very exothermic, and it also produces more neutrons which then fly off and break apart more nuclei producing more energy. However, the density of neutrons required to sustain a reaction is very high, so the fissile material must be kept packed together. The point at which the reaction is producing enough energy and neutrons to sustain itself is the critical mass.
Nuclear reactors must sit between too dense and not dense enough while using the waste heat to produce electricity. Various safety mechanisms manage this. Not dense enough and the reaction cannot sustain itself and it fizzles out. Too dense and the reaction runs away, it goes supercritical, and you get a nuclear meltdown... or a bomb.
A nuclear fission bomb is, basically, a deliberate nuclear meltdown. It's a way to very precisely smash hunks of sub-critical fissile material together to make a critical mass; a prompt criticality. This must happen very precisely because as soon as some fission starts energy will be produced which will vaporize material raising pressure rapidly. This can shove the fissile pieces apart shutting down the reaction before much fission has happened.
The most basic fission bomb is a gun-type which literally shoots a sub-critical pellet of uranium into a sub-critical cylinder of uranium making a critical mass. But it's also very inefficient since as soon as fission starts the uranium blows itself apart stopping the fission. Most of the uranium is never used. Little Boy was a gun-type bomb.
A more efficient, and much more complex, fission bomb is the implosion type. This uses very, very, very precise conventional explosives arranged around many pieces of sub-critical material to simultaneously crush them into a sphere. The force and precision of the conventional explosion holds the fissile material together in a super-critical state for as long as possible to fizz as much material as possible making them very efficient. Fat Man was an implosion-type bomb.
Fusion bombs work on basically the same principle: you need to squeeze the material together very quickly and very precisely before it blows itself apart. They do this with a fission bomb. A fusion ("hydrogen") bomb is basically a conventional bomb that sets off a fission bomb which sets off a fusion bomb.
Now we can see why a nuclear bomb will not set off a nearby stockpile of nuclear material or bombs. An insignificant fraction will fizz because of the particle bombardment, but the blast will simply blow the extra material away before it can achieve critical mass.
If you were to take two pieces of sub-critical fissile and smack them together with your hands to make a critical mass, it would be very bad for you and everyone nearby, but it would not cause a nuclear blast. As Wikipedia dryly puts it...
The prompt-critical excursion is characterized by a power history with an initial prompt-critical spike as previously noted, which either self-terminates or continues with a tail region that decreases over an extended period of time.
In layman's terms: the two halves would blow themselves apart.
This is known as a "criticality accident" or "critical power excursion". This happened at Los Alamos twice when experimenting with the "Demon Core" in terrifyingly unsafe manners; though in both cases the scientists shut down the reactions before they blew themselves apart.
And this is why nuclear weapons are considered "safe". Unlike conventional explosives which can be detonated by a simple fire, a nuclear bomb must work perfectly to go off. This is why they are often referred to as a "device". Any damage to the bomb makes it safer. Safety mechanisms effectively remove critical parts of the device; like how one can ensure a car will not start by pulling out the carburetor or the fuel pump. Arming a nuclear bomb basically finishes putting the device together.
The worst that is likely to happen to an unarmed nuclear device is the conventional explosives will detonate scattering fissile material into the environment. While that's indeed very bad it's much better than a nuclear explosion. We're very sure of this because it has happened a very distressing number of times.