Refers generally to a property of a statistical procedure to go to the "right" place as the sample size tends to infinity, primarily referring to estimators converging to the true parameter value as the sample sizes diverges. Use also for Fisher consistency, the property that an estimator when applied to the complete population gives the right answer.
Consider an iid sample $X = X_1,...,X_n$ and let $T_n(X)$ be a point estimator of some quantity of interest $\theta$, then consistency means that the estimator is consistent if it converges in probability to the true population value as the sample size tends towards infinity: $$T_n(X) \overset{p}{\to} \theta$$
Whilst consistency is a desirable property, an estimator that converges to the true population value at a low rate is not very useful in practice. Therefore it is interesting to know the speed of convergence if this can be determined explicitly. For example, the OLS estimator in the usual cross-section context is $\sqrt{n}$-consistent, meaning that it converges to the population quantities at $\sqrt{n}$ as the sample size increases.
Often this is the highest speed which can be achieved by an estimator, though there are some instances when faster convergence is possible. Such estimators are called super-consistent. An example is again OLS in the context of estimating the long-run relationship between two co-integrated variables as in
$$y_t = \delta_0 + \delta_1 x_t + u_t$$
for which OLS can be shown to converge at rate $T$ (rather than $\sqrt{T}$).
