A thermobaric weapon is a weapon which utilizes the heat (thermos) and pressure (baros) from an explosion to cause a long-duration shockwave which, in turn, damages the target. This is in contrast to high-explosive weapons which produce very sharp but very short-duration overpressure waves. Fuel-air explosives are in fact thermobaric weapons. The term has recently come into vogue in the media, however, to describe new families of solid-fueled weapons which have been weaponized into packages small enough to be fired from man-portable rocket launchers or even personal grenade launchers. While 'normal' fuel-air weapons such as the BLU-73, BLU-95, BLU-96 and the CBU-55 and CBU-72 are usually deployed from aircraft against outside targets or areas, these newer small weapons have found a niche in their use against specific structures or in enclosed spaces. The XM1060 40mm grenade, designed by Picatinny Arsenal for use in Afghanistan by the U.S. Armed Forces is fired from the M-79 Grenade Launcher or the rifle-attached M203 Grenade Launcher and has been employed with great success against enemy strongpoints in buildings and caves where direct fire and even fragmenting grenade rounds have been ineffective.
The reason these rounds are so effective against sheltered enemy positions is that unlike direct fire or fragmentation rounds, there need not be a clear line of sight beween the detonation of the weapon and the target. Inside buildings, bunkers or caves, the overpressure wave that results from the explosion will cause damage anywhere that air can reach from the explosion point. Due to the long duration of the overpressure wave, if the target is inside a relatively weak structure there is a much higher probability that the structure itself will be damaged, perhaps terminally.
Thermobaric weapons operate as a fast sequence of events. They consist of a fuel (usually a powdered solid, in the case of the smaller weapons; or a fluid in the case of many of the larger fuel-air explosive weapons) which is contained within the weapon's shell. There is are one or more high explosive dispersal charges inside the weapon as well, and perhaps an ignitor. When the round is detonated, the high explosive dispersal charge (also known as a bursting charge) splits open the weapon case and disperses the fuel over a wide volume. It is powdered finely enough (if solid) or aerosolizes finely enough (if liquid) to remain suspended in the nearby atmosphere. After a time, the ignitor squib fires. The fuel, now mixed with local oxygen, deflagrates. This combustion puts off enough heat and high-pressure waste products to form a general high pressure heated wave which travels outward.
Some thermobaric weapons (the smaller ones, typically) do not need a separate ignitor system. Their bursting charge ignites the particles closest to the charge itself even as it disperses the full load; the burning particles form enough of a hot pressure wave to ignite those particles further away when it overtakes them. This lack of a requirement for a separate igniter increases the reliability of the weapon as well as decreasing its cost and weight. In some cases, the particles are of varying size, with the lighter motes burning intially and quickly, creating enough heat to ignite the longer-burning large particles. One advantage of the thermobaric weapon is that if used against a target containing flammables (explosives, ammunition, etc.) it is much more likely to destroy them due to its volumetric flash heating and shock.
filoraene points out quite correctly that one of the reasons thermobarics are so dangerous is that the energy density of their fuel is higher than the energy density of high explosives. Thus the critical characteristic of a thermobaric weapon is its ability to utilize more energy, released over a longer time period, for its lethal effects.
There has been a large argument over the 'ethics' of the deployment of these weapons. In most cases, with the smaller weapons, the actual lethality of the weapon is increased within its range of effect, but the mechanism of damage has changed from shrapnel to overpressure. In both cases, however, the weapon produces mechanical effects. Thermobaric weapons are usually not dangerous if they misfire; unlike cluster bombs, they will leave a fine spray of metal powder (typically) over the area. The larger bombs use fuels which are toxic before combustion, meaning misfiring large fuel-air explosives can stray towards chemical weapon effects. However, even there, they are not as bad, since their fuels are designed to disperse into the atmosphere rather than remain persistent as chemical weapons are.