Between-subjects effect becomes non-signicant after centering covariate. Should I center or not? I've used a general linear model function  to run an ANCOVA, involving a categorical predictor (with two levels), a continuous predictor, and their interaction effect. 
If I don't center the continuous predictor, I get a significant effect for the categorical predictor and a significant interaction effect. If I mean-center the covariate, the categorical predictor effect becomes non-significant but the other effects remain unaltered. Does anyone know why this happens? And which option is the correct one (to center or not to center?)? 
 A: Neither, or both, are correct.
You often cannot interpret the main effect when you have an interaction in there. The main effect is the difference between the groups when the covariate is equal to zero - if zero is meaningful, then it's interpretable. If you move where 0 is (by centering) you change the main effect of the between subjects factor. But the actual model doesn't change, you just shuffle the parameters about. 
Usually a graph is sufficient to see what's going on. If you want to get more technical, you can calculate 'regions of significance', see: http://quantpsy.org/interact/
(Note that ANCOVA is multiple regression, which is why the pages refer to multiple regression.)

Here's a picture, the lines of best fit for two groups are shown. If the intercept is at the left, then group 2 has a higher score. If the intercept is on the right, then group 1 has the higher score. If you center the covariate, then the two lines are at the same height, and there's no difference. 
A: In your first regression the mean of the covariate is in both covariate itself and the intercept, and both are significant. In the regression notice what happens to the intercept: it must have changed quite a bit, because now the covariate's mean went into it.
UPDATE
For instance, say the true process is $y=x+\varepsilon$, but you model it as $y=\beta_0+\beta_1z+\epsilon$ where $z=x-1$. If you estimate the model, the intercept should come as $\beta_0=1$, because $y=\beta_0-\beta_1+\beta_1x+\epsilon\equiv x+\varepsilon$ which will make $\beta_1=1$ and $\beta_0-1=1$
On the other hand, if you estimate the "correct" model, i.e. $y=\beta_0'+\beta_1'x+\varepsilon$, then the intercept must come as $\beta_0'=0$.
Concluding, I think the interaction is a red herring in this case. The real phenomenon is that when you subtract the mean from the main effects it will go into the intercept. Depending on the situation it may or may not change the significance of the intercept. It will definitely change its estimated value.
