Decision rule

A decision rule is the verdict written down before the data exist: fix $\alpha$ (alpha, the significance level), then reject $H_0$ when the p-value is at or below $\alpha$, and fail to reject otherwise. Fixing it first is what gives $\alpha$ any meaning. $\alpha$ is the long-run probability of rejecting a null hypothesis that is true, and that guarantee attaches to a rule chosen without reference to the result.

At $\alpha = 0.05$, a p-value of 0.032 rejects and a p-value of 0.080 does not. The comparison is at or below, so a p-value of exactly 0.050 rejects. Notice how little the data have to do with it: that same 0.080 rejects at $\alpha = 0.10$, and that same 0.032 fails to reject at $\alpha = 0.01$. The rule, not the evidence, is what changed.

"We accept the null hypothesis" is not one of the two outputs. Failing to reject means the data would not be surprising if $H_0$ were true, and that is equally true of a whole range of parameter values near it, which is what a confidence interval displays. The quieter version of the same offence is "$p = 0.06$ was almost significant, so we reject at $\alpha = 0.10$." Moving the threshold after seeing the p-value converts a 5 percent rule into a 10 percent one and throws away the guarantee $\alpha$ was carrying.

The rule returns a verdict, not a strength, and it says nothing about whether this particular call was correct. $\alpha$ controls how often true nulls get rejected across many tests; the single test in front of you either got it right or committed a Type I error, and no rule can tell you which. That is why choosing $\alpha$ is really a judgment about which mistake costs more, since lowering it trades Type I errors for Type II errors.

Comparing the test statistic against a critical value instead is the same rule in different clothing. The statistic falls in the rejection region exactly when the p-value is at or below $\alpha$, so the two routes cannot disagree.