A finite sample breakdown point is that fraction of the data that can be given arbitrary values without making the estimator go bad. This is however, a local breakdown point.
The asymptotic breakdown point is the limit of the finite sample breakdown point as n tends to infinity. Greater the value of the breakdown point better is the estimator used.
For example, suppose for a data set there are three estimators: mean, pseudomedian and median. The breakdown point for mean is 0, pseudomedian is 0.293 and median is 0.5, then the sample mean is the worst estimator, the pseudomedian and median are better, and since the median estimator has the highest breakdown point it is the best estimator. The amount of robustness required also depends on the strength of the data used.