In hypothesis testing, the alpha value is the probability of making a Type I error. This is the type of error mentioned in the question.
Type I error - you find a difference when a difference doens't really exist.
One way of remembering this is that this is what scientists "want" to make: they want to find a significant difference in their data, thus it is the "first mistake" they'd make.
Alpha is the probability you are willing to accept that you could have made a type I error (i.e. an alpha value of 0.05 means there is a 5% probability you could make a type I error and reject the null hypothesis when you should not)
Type II error - you do not find a difference when you should have because a true difference really exists
Beta is the probability that you make a type II error
Power is equal to (1 - beta)
Power can be increased by increasing sample size, and thus with a larger sample you have a lower probability of making a Type II error
Power can also be increased by increasing expected effect size or increasing precision. It is interesting to note accuracy has no effect on power.