Alpha is denoted for the significance level in null hypothesis. It is mainly the probability of rejecting the null hypothesis when true. As far as I know, it is the highest value that we are willing to accept making mistakes.
See, the point is that when we do hypotheses testing in statistics, we always try to negate the opposite hypothesis i.e. null hypothesis but at the same time it’s next to impossible to affirm the hypothesis we are interested in. and we through our observations and research, null hypothesis can be specified easily but different is the matter with the other.
No, it never means that the null hypothesis is true if you are unable to reject hypothesis and false is the alternative. Lack of proofs, evidences and data can be one of the reasons for such failure.