Definition: An estimator $\bth$ is said to be a consistent estimator of $\bt$ if $\bth \inP \bt$.
The idea of consistency is often too weak to be interesting: any reasonable estimator is consistent given an infinite amount of data. A stronger idea is $\sqrt{n}$-consistency.
Definition: An estimator $\bth$ is said to be a $\sqrt{n}$-consistent estimator of $\bt$ if $\norm{\bth-\bt} = O_p(1/\sqrt{n})$.
This means that not only does $\bth$ get close to $\bt$ as $n \to \infty$, but it converges to $\bt$ so fast that it stays within an ever-shrinking neighborhood $N_{M/\sqrt{n}}(\bt)$ with high probability.
See also: Proving consistency via quadratic mean