For an unbiased estimator, asymptotic efficiency is the limit of its efficiency as the sample size tends to infinity. An estimator with asymptotic efficiency 1.0 is said to be an "asymptotically efficient estimator". Roughly speaking, the precision of an asymptotically efficient estimator tends to the theoretical limit as the sample size grows.
Asymptotic efficiency of an estimator depends on the population. For one sort of populations (distributions) an estimator may be asymptotically efficient, for others - not asymptotically efficient.
Among known estimators, the number of asymptotically efficient estimators is much greater than the number of efficient estimators.
Note: asymptotic efficiency (of an estimator) should not be confused with asymptotic relative efficiency (of two estimators).