Suppose I have two experiments A and B. Each experiment is composed of N elements that are clearly separated into two groups. In experiment B one of the groups has a higher mean than the same group in experiment A. This difference (shown in red in the image) is what I am trying to understand. I need to show that it is significant, or not, determined by any value. p-value? a test?
To show statistical significance (that is, to show that the results of your experiments are not just due to chance), you first need to define your hypotheses (the null and the alternative hypothesis). In this case, the null hypothesis, $H_0$, could be "There is no difference between the mean of the groups in experiments $A$ and $B$". The alternative hypothesis, $H_a$, could be "There is a difference".
You now perform a statistical test. If the resulting p-value (returned by the statistical test) is less than a threshold value (often called the "significance level" and it is e.g. $0.05$), then you reject the null hypothesis and you say that this result (that is, the p-value less than the significance level) is statistical significant.
If the distribution is not too different form normal and variances intra-group equal, an ordinary t-test will do (function t.test in R, for instance). If nothing can be assumed about the distribution of both groups, you might use a permutation test --just google for "permutation test R" to obtain lots of pointers.