The ballistic coefficient β is a measurement of an object's ability to move through a fluid. It takes into account the effects of an object's density and its skin friction, and is calculated as follows:
β = m * CD / A
β is typically expressed in kg/m2 . I've never seen it expressed in English units, but I imagine it's expressed in psi. Quite silly. Of course, there is also the unitless measurement of beta, as compared to the G1 standard bullet, used exclusively in the realm of firearms. In that case, β is expressed as a multiple or fraction of the drag of a G1 Bullet. In most cases, higher numbers are better.
The ballistic coefficient describes how closely an object will follow a ballistic trajectory in a fluid (typically air). Ballistic coefficient should be given with a specific speed, since this is another factor which changes the object's ability to slice cleanly through the fluid, and is not included in measurements of β. Typically, a reference will be to "beta at Mach 1, standard atmosphere." Since CD (a component of beta) depends on the fluid density, this removes all ambiguity from the measurement.
Objects for which a high ballistic coefficient is important are bullets, artillery shells, and reentry vehicles. For bullets and objects designed to fly horizontally, higher ballistic coefficient means less vulnerability to drag. The more drag acts on the bullet, the longer it's in the air, and the longer extra forces like gravity act on it, steering it off-course. For a reentry vehicle or other projectile already using gravity to get to its target, the most important factor in hitting the target is maintaining velocity after it's aimed; a lower ballistic coefficient means a slower impact speed and a longer flight, as well as significantly reduced range and accuracy.
For manned spacecraft (think Apollo capsules), accuracy and insufficient velocity aren't problems, but the heat and excessive impact speed are. The higher the capsule's beta, the faster it comes in through the high atmosphere. This means that the heat encountered during reentry comes all at once, in a super-high spike, and that the capsule could still be moving supersonically when it hits the ground (yes, hits the ground... with people inside... at Mach-something-bad). Needless to say, this is not a friendly environment in which to deploy parachutes. If, however, the capsule is designed to be blunt, and have a large bottom surface area, its beta will be low, and more kinetic energy will be bled off as heat. This translates to a lower peak temperature, a higher transition to subsonic speeds, for the cost of a longer heating envelope.
A final note for any firearms enthusiasts: just as you shouldn't buy a computer based only on gigahertz, don't make ammunition purchase choices based solely on beta. Many companies are still wavering on whether the G1 standard bullet is a good standard, since in many cases G2, G5-G8, and GI can each be a better benchmark for a specific bullet shape. Ask questions and read up. And for you missile enthusiasts out there: the same advice holds true. C'mon... a nuclear-armed reentry vehicle doesn't need to be that accurate.