73 Actually, nukes are set off by conventional explosives. Not sure if they mostly used gunpowder or high explosives; might depend on the nuke.
For uranium-based nukes like the "Little Boy" bomb dropped on Hiroshima, there are two hemispheres of weapons-grade uranium just below critical mass situated at either end of a metal tube. Behind one is an explosive charge, probably based on gunpowder. When the charge is set off, it shoots that uranium hemisphere down the tube where it runs into the other one, effectively creating one larger lump of uranium well above critical mass that then explodes.
For plutonium, that gun-like design just doesn't work. Instead, they use a design similar to the Fat Man nuke dropped on Nagasaki. In the core of these nukes, there is only one solid lump of fissile material, once again just below critical mass. This is completely surrounded by conventional explosives (I'm guessing C-4 or TNT), all encased within the nuke's metal shell. A blasting cap sets off the TNT, which detonates, squeezing the lump of plutonium until it's just above critical mass, so it explodes.
Thermonuclear weapons are... more complicated. The gist of it is there's a small container of deuterium and tritium buried somewhere in the middle of the nuke, and there's a bunch of fissile material surrounding it, so when the shell gets set off by a Fat-Man-style nuke right next to it, it produces a flood of radiation that heats and compresses the D-T payload, creating conditions similar to those in the core of the Sun and then fusion and boom.
Some even throw in another chunk of nuclear waste or something that can be activated by the flood of neutrons from the fusion reaction, which just makes it even boomier.