Post hoc tests
Once a significant F-value is obtained in an Analysis of Variance, your work is not yet over. A significant F-value tells you only that the means are not all equal (i.e., reject the null hypothesis). You still do not know exactly which means are significantly different from which other ones. You need to examine the numbers more carefully to be able to say exactly where the significant differences are. In the example above, the significant F-value would allow us to conclude that the smallest and largest means were significant different from each other, but what about Mean 2 and Mean 3 or Mean 2 and Mean 4? Hence we need post hoc tests.
The most widely used post hoc test in Psychology and the behavioural sciences is Tukey's Honestly Significant Difference or HSD test. There are many types of post hoc tests all based on different assumptions and for different purposes. Tukey's HSD is a versatile, easily calculated technique that allows you to answer just about any follow up question you may have from the ANOVA.
Post hoc tests can only be used when the 'omnibus' ANOVA found a significant effect. If the F-value for a factor turns out nonsignificant, you cannot go further with the analysis. This 'protects' the post hoc test from being (ab)used too liberally. They are designed to keep the experimentwise error rate to acceptable levels.