The neutrons released in fission can then hit another nucleus and trigger another fission, and what happens next depends on the amount and arrangement of fissile material present and on any other materials that are around. If the arrangement is such that the neutrons from fission produce, on average, exactly one more neutron then the number of nuclei that undergo fission is constant in time and the reactor is described as critical. If, on average, each neutron produces less than one more neutron, the reactor is sub-critical and, in the absence of any external source of neutrons, the number of neutrons decreases exponentially and the reactor very quickly ceases to produce energy. If, on the other hand, each neutron leads to more than one more neutron, the reactor is supercritical and the amount of energy produced will increase exponentially until either some process or control system alters the neutron multiplication factor to bring it back under one, or until the reactor melts or explodes. Normal reactors are operated at criticality, on the knife-edge between sub-critical shutdown and supercritical runaway, with a number of inherent processes and control systems constantly adjusting the neutron multiplication factor to keep it precisely tuned to one.
An ADSR differs from an ordinary reactor in two critical aspects — the first being that it isn't critical, in fact it is intentionally designed to always be sub-critical. This means that left to itself it immediately shuts itself off. That would be very safe, but pretty much useless, without the other difference — an ADSR has an extra source of neutrons that keeps it running in the absence of enough neutrons from fission. These extra neutrons are supplied by having a particle accelerator that can collide a beam of energetic protons with a target in the centre of the reactor. These collisions produce neutrons by a process called spallation, and these neutrons are then multiplied up in number by fissions in the reactor core, resulting in the desired power level.
An ADSR is obviously more complicated than a reactor because of the need also to build the high-current, high-reliability accelerator. However, this added complexity buys a number of potential major advantages. First, the sort of runaway criticality excursion that blew up the Chernobyl reactor is essentially impossible for an ADSR, since they run very far from criticality. Second, the relaxation of the need to balance the neutron economy so finely means that an ADSR could run with a wider range of potential nuclear fuels. In particular, the extra neutrons produced in an ADSR could be used to convert non-fissile thorium-232, which has such a large natural abundance that we would have fuel for many centuries, into fissile uranium-233, which could then fuel the reactor. The extra neutrons can also potentially clean up another problem, which is the long-lived nuclear waste produced by current reactors (and which would be produced, although in much smaller quantities, by a thorium-fuelled ADSR). The long-lived component of the waste is mostly the plutonium and other similar elements called minor actinides. These minor actinides can also be made to undergo fission, so they can be separated and burned up in the ADSR, greatly reducing the long-term radiation hazard from the waste. Another major (and underappreciated) advantage of an ADSR is that the power output is controllable over a wide range just by changing the current from the particle accelerator. A conventional reactor usually runs efficiently over only within a small range of output power, which makes it hard to match power production to consumption if most of your power comes from such reactors — a set of ADSRs would provide the perfect source for tunable power to deal with fluctuations in demand.
What about proliferation? Here the answer is a bit subtle. There are basically two materials used to build fission bombs today. First, there is highly-enriched uranium, or HEU, where the uranium-235 has been separated from the far more abundant uranium-238. This is the extremely tricky process that Iran is trying to master. However, if you can do the separation and make HEU, converting that into a nuclear weapon is actually quite easy. The other option is to use the uranium in a reactor, which will transmute some fraction of it into plutonium-239. This can be chemically separated from the parent uranium, so it is rather easier to get your hands on plutonium-239 than HEU. However, the compensating disadvantage is that it is much more technically challenging to make an effective nuclear weapon out of plutonium-239 (which is no doubt why the North Koreans were able only to produce a damp squib of an explosion with their first try). A thorium-cycle ADSR nicely avoids both of these weapons materials. First, it doesn't require any HEU, so there is no danger of leakage of uranium-235 from a power programme into a weapons programme. Second, it does not produce significant quantities of plutonium (in fact it would probably be used to consume plutonium, not produce it), so it does not lead to the ability to build that type of bomb either. The one fly in the ointment is that it does produce large quantities of uranium-233, which is the fissile material that actually produces the power from the reactor. Now there are various reasons why this uranium-233 would be less desirable for use in any weapons programme (for one thing, if made in an ADSR it would inevitably contain very intensely radioactive uranium-232, making any resulting bomb hard to build, hard to store, and extremely hard to conceal). However, you could not say from first principles that you could not possibly build a bomb out of it, so I don't think you can say that you cannot build a bomb based on an ADSR programme — only that it would hugely complicate the task of making a nuclear weapon starting from an energy programme.
