When one ordinarily thinks about
nuclear reactors, one probably imagines
fuel rods,
control rods, a
heat exchange,
cooling towers, and so forth -- not the kind of assemblage one can just shoot into space, due if for no other reason to their raw tonnage. RTGs, however, are low output, no-moving-parts nuclear reactors that can be as small as, say, a two-drawer
filing cabinet, or even a
grapefruit. They've been in use for various kinds space missions since roughly 1961, and one might speculate that the first production model was in testing a few years beforehand (though since the concepts involved are so simple there was likely experimentation long before). Recently people have been paying more attention to these because of the possibility of their fuel supplies being dispersed in the
atmosphere upon accidental satellite re-entry.
Crucial to the operation of an RTG is the Seebeck effect, the physical counterpart to the better-known Peltier effect. In a Peltier setup current is passed through two sheets of different metals or semiconductors pressed against each other, causing one side to become warmer and the other cooler. A Seebeck setup instead relies on the temperature gradient between Plutonium -- which gets pretty hot, +700 degrees K during its radioactive decay -- and an external heat sink. Between the Pu and the heat sink are two plates as in the Peltier setup, and current is thus generated as the heat moves through the them.
Generally, and in the example above, Plutonium-238 -- not a weapons grade substance -- is used for United States RTGs, and the USSR used enriched Uranium-235 and Polonium-210 for some of theirs. Plutonium-238 is the longest lived of the bunch, with a half-life of 87.7 years, where the Polonium-210 goes through a half life in a mere 138 days. Because of the continuous nature of Seebeck effect power generation, the power output falls off gracefully from the reactors, so low-power utilities on a satellite may still be usable even when high-power ones are not. Power output can be as low as the designer requires, up to about 1000 watts. For reference, the reactor sent up in 1961 produced 2.7 watts at its outset, and each of the two RTGs on the Galileo spacecraft produced about 285 watts.
Modern RTGs are designed in such a way that even with an unplanned re-entry the Plutonium or other material would (hopefully) not be dispersed in the atmosphere. Instead of a solid rod of fuel, which would be a terrible idea with Plutonium anyway, it is separated into many cylindrical pellets, each of which are first clad with Iridium, then layered with graphite. Upon re-entry the graphite is meant to burn away and leave enough behind to absorb landing impact, hopefully leaving the "bullet" of plutonium whole. Also, the plutonium is made into a ceramic rather than left as a pure metal, with the hope of additional re-entry heat absorption and stability against atomization.
Notably, most of the radiation released from Plutonium-238 during decay is in the form of alpha particles, which are absorbed or reflected by the iridium cladding. This is why the RTG set up on a moon walk during the Apollo 16 mission was able to have its fuel rod installed by somebody wearing a space suit, without that person, um, dying. There is also some neutron and gamma wave output; since those particles are difficult to shield, satellite RTGs are usually kept spatially separate from the spacecraft's body, at the end of a 5 or 10 meter boom.