What does it mean a value is statistically significant and when does it occur? For example, at a 5% level of significance, do p-values above or below 0.05 indicate significance?
Statistically significant means that the observed value and population value are significantly different.
For Example:-you observed a sample mean $\bar x$ and you check if its significantly different from population mean $\mu$. You can test by Z-test or else. the statistical tests tell you whether the difference is Random in nature(i.e. under the confidence interval) or its Statistically significant.
If the sway between observed value and population value is statistically large enough, then that is statistically significant.
P-value:- It is the minimum level where value is statistically significant(you can reject $H_0$)
So, reject $H_0$ if p-value less than or equal to level of significance.