# when to reject a hypothesis, size and p-value

When explaining the reason for rejecting a null hypothesis, I sometimes see "p-value is small", and sometimes it's "for a test of size $$\alpha$$, we reject it when (some condition)". I was wondering if "p-value is small" is a special case of a commonly used/standard (idk how to phrase this) size $$\alpha$$ test?
If this is not the case, could you explain when I am supposed to use which?
Thank you!

We set some value, called $$\alpha$$, as our maximum tolerance for type I error rate. That is, we accept that our work could reject true null hypotheses $$100\alpha\%$$ of the time the null hypothesis is true. In the common situation of $$\alpha=0.05$$, we accept that to be $$5\%$$. In fact, $$\alpha=0.05$$ is so common that it typically is implied when no $$\alpha$$ is specified, and we consider p-values of $$0.05$$ or smaller to be “small” p-values.
Then we run the test and calculate a p-value. If $$p\le\alpha$$, we reject the null hypothesis in favor of the alternative hypothesis.