P-value can be interpreted as the observed chance of committing a false positive error. Therefore, if the chance is sufficiently low, we may reject the null hypothesis without being likely to be wrong. The usual threshold is 0.05.
With today's advanced computing facilities, p-value can often be obtained from a statistical software. However, we need to be clear about the null and the alternative hypotheses before we can properly draw our conclusions.
What does a p-value of 0.03 mean? Of course it is a statistically significant result at 5% level of significance. But we need to know the null and the alternative hypotheses before we can draw the correct conclusion. In a t-test on H0: mean blood pressure >= 120 against HA: mean blood pressure < 120 mmHg, a p-value = 0.03 means we have sufficient evidence to demonstrate the mean blood pressure was below 120 mmHg. However, in a t-test on H0: mean blood pressure <= 120 against HA: mean blood pressure > 120 mmHg, the interpretation will be quite different.
What does a p-value of 0.56 mean? Of course, it is statistically insignificant at any reasonably level of significance.
In a Chi-square test for association, it means we do not have sufficient evidence to show the association at say 5% level of significance. Note however we should not say there is no association since we will almost surely reach a significant result as the sample size becomes large.