Significant F in ANOVA What does a significant F statistic in ANOVA mean about the differences among the three or more groups means being compared in the test
 A: Sounds like you are doing a one-way ANOVA. 
Formally it means that - if the population means were in fact not different and teh other assumptions hold - an F statistic at least as large as the one observed would be unlikely to be observed (specifically, with lower probability than the chosen significance level).
Somewhat less formally, assuming your significance level is small, then to get a significant result one of three things happened:
1) All assumptions hold, the population means don't differ and you just happened to get an unusually large F value by chance. Since probability of that chance event is quite low, this would be surprising*.  
* At typical significance levels, perhaps only mildly so.
2) All assumptions hold, and the population means differ. A large F statistic is expected rather than surprising.
3) The assumptions fail -- in a way that makes a large F value less surprising.
(well, there's some other possibilities, such as "the population means are equal, and the assumptions fail in a way that makes a large F value even rarer than than when they do hold, but an incredibly unlikely event occurred anyway, resulting in a large F", but the above three are the main ones)
Since you already assumed (3) is not the case, you're left to decide between (1) and (2), and that's why you choose the significance level to be some small probability - it's the probability at which (1) becomes just unlikely/surprising enough that you think (2) is a more reasonable explanation.
A: The $F$ stat is useful for deciding what factors to include in a model, but then it doesn't have much use in helping understand what's really going on because it doesn't tell you anything about the pattern of the means. But that's ok, because you can actually look at the means and find out what that pattern is. So you should plot them, and do some multiple comparisons of the means, using, say, the Tukey method, and find out which means differ significantly. 
