Strong, persistent correlation between A and B generally implies that it's one of three scenarios: A->B, B->A, or C->A + C->B.
In cases where it's not possible to do proper experiments to explicitly test A->B hypothesis, you'd say that correlation implies causation in the (not that rare) scenario where you have strong reasons to believe that B->A is not possible (e.g. the increase in teen suicides IMHO did not cause the increase in screen time) and you have thoroughly went through all the plausible confounders (C's) and have argumentation/evidence why that's not the case. In that case it's still not solid proof, but it's certainly a good implication and worth considering as likely true until further analysis or evidence.
Effects measured as part of a randomization procedure in a controlled experiment or, failing that, a causal inference model applied to observational data.
Many of the answers here are pointing to methods, but I take this question as more fundamental.
Epistemologically, causation is the intersection of three things:
1) Temporal precedence
2) Covariance (i.e. correlation)
3) Absence of likely alternative explanations
Number 3 is by far the trickiest. Our inability to definitively rule out all possible alternatives means one has to resort to inference where causation is concerned.
Randomised intervention - i.e. randomly selecting a group of people to change their behaviour and comparing to the group that doesn’t change, who ideally receives a placebo. The random part is important, any other type of selection (e.g. observing people who choose to change their own behaviour) doesn’t work, in theory at least.