Taken loosely, sensitivity means the ability to respond to something if it's present, and specificity means the ability to suppress responding when it's absent. For continuous variables, sensitivity corresponds to the slope of the regression of the obtained measures on the true values of the variable being measured, and specificity corresponds to the standard error of measurement (i.e., the standard deviation of the obtained measures when the quantity being measured does not vary).
EDIT, responding to comments by Frank Harrell and cbeleites. I was trying to give conceptual analogs of sensitivity and specificity. For continuous variables, the basic idea of sensitivity is that if two objects (or the same object at different times or under different conditions, etc) differ on the variable we are trying to measure, then our obtained measures should also differ, with bigger true differences leading to bigger measured differences.
The regression of any variable, say $Y$, on any other, say $X$, is simply the conditional expected value, $\mathrm{E}\,Y|X$, treated as a function of $X$. The sensitivity of $Y$ to $X$ is the slope of that function -- i.e., its derivative with respect to $X$ -- evaluated at whatever values of $X$ are of interest, and possibly averaged with weights that reflect the relative importance or frequency of occurrence of different $X$-values.
The basic idea of specificity is the converse of sensitivity: if $Y$ has high specificity and there are no true differences on $X$ then all our measured $Y$-values should be the same, regardless of whatever differences there may be among the objects on variables other than $X$; $Y$ should not respond to those differences. When $X$ is constant, higher variability among the $Y$-values implies lower specificity. The conditional standard deviation -- i.e., the s.d. of $Y|X$, again treated as a function of $X$ -- is an inverse measure of specificity. The ratio of the conditional slope over the conditional s.d. is a signal-to-noise ratio, and its square is referred to in psychometrics as the information function.