If two statistical procedures have the same limit as $n_1 \to \infty$ and $n_2 \to \infty$, then the asymptotic relative efficiency (ARE) is the limit of the ratio $n_1/n_2$. More specific definitions, as applied to the special cases of estimation and testing, are given below.
Estimators
Definition: If $\sqrt{n}(\th_1-\theta) \inD \Norm(0, \sigma_1^2)$ and $\sqrt{n}(\th_2-\theta) \inD \Norm(0, \sigma_2^2)$, the asymptotic relative efficiency (ARE) of the two estimators is $\sigma_1^2/\sigma_2^2$.
Tests
Consider testing the null hypothesis $H_0: \theta = \theta_0$ against the sequence of alternative hypotheses $H_a: \theta = \theta_0 + \Delta/\sqrt{n}$. If $\beta_1 \to \Phi(\Delta a_1 - z_{(1-\alpha)})$ and $\beta_2 \to \Phi(\Delta a_2 - z_{(1-\alpha)})$, where $\beta_i$ is the power of test $i$, the asymptotic relative efficiency of the two tests is $(a_1/a_2)^2$.