The first nuclear bombs were very simple, they had a big sphere of fissionable material (uranium or plutonium) with a plug cut out, and a sort of gun barrel to fire the plug into the sphere. The sphere-with-a-hole was subcritical, but adding the plug would take it over critical mass and start a chain reaction, which leads to a nuclear explosion. The trigger doesn’t have to be real exact on one of these, just something that slams the chunk into the sphere. They weighed in at something like 10 tons apiece, so weren’t exactly suitcase-nukes.
Modern nuclear weapons take advantage of the fact that ‘critical mass’ is more accurately ‘critical density’, and use a carefully constructed pattern of conventional explosives to crush a much smaller chunk of fissionable material into a dense enough ball for a chain reaction. This requires a lot less fissile material, but is also a lot harder to do - the conventional explosives need to be set off in a very exact pattern, and finding exactly how to set up and trigger the explosives is complicated work (and extremely secret). They can be much smaller, though ‘suticase nukes’ are still pretty hard to make.
http://www.fas.org/ has a lot of information on nuclear weapons, treaties related to them, and who probably has them.