The power, or sensitivity, of a one-way ANOVA is the probability that the test will be able to detect a difference among the groups (i.e. reach statistical significance) when there really is a difference.
Estimate the power of your test while designing your experiment to select sample sizes appropriate to detect an amount of difference between means that you deem important.
Too small a sample is a waste of experiment, but too large a sample is also a waste of resources.
A power of at least 80% is often suggested.