Damage Expectancy is a measurement used in the targeting of weapons and the tasking of assets. There are two general uses for the term; one deals with nuclear weapons, and the other with conventional weapons. Really, there is no qualitative difference, simply a matter of scale; however, the processes of tasking and targeting the two types of weapons are quite different, and thus the uses are considered distinct.

But what is it?
Good question. Let's start with the following definition, from the U.S. Department of Defense Technical Information Center (http://www.dtic.mil):

The probability that a weapon will arrive, detonate, and achieve at least a specified level of damage (severe or moderate) against a given target. Damage expectancy is a function of both probability of arrival and probability of damage of a weapon.

Note that this is the explicitly nuclear use of the weapon. Here, we are told that in order to determine the damage expectancy (DE) of a weapon against a target, we must consider in essence everything that happens from the launch of the weapon to its detonation at or near the target. The DE is the chance that a single weapon will traverse that entire timespan and succeed in damaging the target. This calculation will, in the case of a nuclear attack, subsume factors ranging from the reliability of the launcher to the effectiveness of any defensive systems protecting the target, as well as the CEP of the weapon and the hardness of the target itself.

The definition above notes that DE should be expressed in a qualitative estimate of damage to the target. While this may be true for deciding how badly a constant-function target (such as, say, a city or railyard, or marshaling area) may be damaged, it is not used when planning counterforce strikes - strikes against point, high-value time-dependent targets. Typically, this means against your enemy's own nuclear weapons. You don't care, in this case, if his silo has been 'moderately damaged' - you care if it will work or not. In such cases, DE is usually expressed as a percentage or a probability; in sum, the probability that each weapon launched will succeed in disabling/destroying the target. This usage is also found in conventional weapons targeting, which is much more familiar with the notion of 'hard' targets and multiple weapons - after all, you have to drop a 500-lb bomb much closer to its target to be sure you destroyed it!

What is it used for?
When deciding what should be targeted, and how it should be attacked, planners will rely on this measurement to determine how many attempts should be made to destroy the target per sortie. This is another area where the 'nuclear' versus 'conventional' uses diverge; in a nuclear exchange, the implicit assumption is that you (the planner) will only have one chance to achieve the desired Probability of Kill (Pk) against a target. In conventional strikes, you'd like to take it out in the first shot, as well; however, it's much more likely you'll be able to schedule follow-on strikes or reconnaissance. In other words, the strike does not happen as a single shot; there will be action, reaction, later chances, etc.