In a
nuclear detonation, prompt
radiation is radiation delivered directly to a
target or other object from the
event itself. This is in contrast to
fallout, which is
contamination of an area or object due to the presence of
radioactive dust, made so by the original blast. Most
fallout is in fact
soil particles tossed into the upper atmosphere by a nuclear explosion which then 'falls out' of the air.
There are many components to prompt radiation. An atomic detonation releases alpha particles, beta particles, neutrons of various energy states, gamma rays, and a hellacious amount of less-energetic photons of varying energy levels, like visible light, infrared and ultraviolet, X-rays, etc. These have differing levels of effects on human and inanimate targets.
Alpha particles, as explained in their node, are essentially helium nuclei which have been stripped of their electrons. They are not terribly energetic, and are stopped by the least substantial of shielding. Human skin, in fact, will block alpha particles without (much) damage.
Beta particles are high-speed electrons, stripped from their atoms and blown outward as plasma. They are harmful to human tissue, but have almost no penetrating power. Exposure to large amounts of beta radiation will typically cause severe damage to the human epidermis' outer layers (think super sunburn); however, such damage will heal in time, and most fatalities from such exposure come from infections contracted while the lower dermis is exposed and vulnerable.
Gamma rays are a real killer, the 'radiation' that people speak of in fear. Gamma rays are extremely high energy photons (I'm sure someone else can tell you their eV ranges), and do severe damage to human tissue. They destroy and disrupt genetic material, leading to mutations and cancers in the medium to long-term, and destroy cells they strike, leading to a more immediate problem.
Neutrons both damage targets directly (they have a nasty effect on human tissue, in large enough quantities and at high enough speeds) but also by causing otherwise stable materials in the target area to become radioactive. A neutron may hit an atom in the target area and be absorbed, leaving the atom unstable and likely to decay in the next 24-48 hours or so. Thus, even if the main radiation wave is avoided, there remains danger from irradiated materials. This is different from fallout.
Paradoxically, the larger the atomic detonation, the less likely you are to suffer from prompt radiation. Really large explosions will tend to destroy all nearby objects through blast effects and secondary effects like ignited fires and debris. Prompt radiation is absorbed by the atmosphere, so beyond a certain distance from the epicenter, it isn't typically a problem. There is also the problem of plasma eclipse. During the very early phases of an atomic explosion, the prompt radiation released (mostly in the X-ray ranges) hits the surrounding atmosphere and heats it to plasma almost immediately. The heated plasma, however, serves to block all other radiation from passing through it for a fraction of a second to several seconds, depending on the size of the blast; this is why there is a 'dark period' during the detonation of an atomic bomb. Eventually, the heated plasma expands (this is the fireball) and in so doing, cools enough to pass radiation once more. The most reliable method of measuring the size of an observed atomic detonation is to precisely time the interval between the two peaks of maximum observed brightness - this is how the bhangmeter works. That period is the plasma eclipse, and since the atmosphere is a relative constant, you can calculate the energy release from the explosion.
In the unhappy event that you witness an atomic explosion, diving for cover is not pointless. The larger the bomb, the longer the time period over which the radiation release is spread (due to the plasma eclipse) and the longer the initial delay. Managing to put a meter or more of earth between you and the blast will go a long, long way towards nullifying the prompt radiation effects; thus, diving behind a wall or building will in fact lessen the effects significantly. This is one reason for the seemingly-asinine 'Duck and Cover!' campaigns of public service announcements during the 1950s in the U.S. For more info, see The Atomic Cafe.
Thanks to rootbeer277 for factual corrections to the above!