P-Value in one vs two sided tests Let's assume we have a right one-sided test with a p-value of 0.03 and a positive test statistical value. Now let's perform a two-sided test with the same datums. Are we going to reject H0?
The significance level is 5%, the test could be either Z or T with one population, so the distribution is symmetric.

The two-sided p-value in the question will be 0.06, insignificant at 5% level, as two probabilities 0.03 have to be added from both sides. But this was apparently clear already.
The more interesting question is, how is it possible and how can it be reasonable that the same data reject the $H_0$ when testing one-sided, but don't reject when testing two-sided?
The reason is that in general the significance level is actually a performance characteristic of the test. The idea is that if you test a lot of times at level 5% in situations in which the null hypothesis is true, 5% of the tests will (wrongly) reject the null hypothesis.
Now if you test one-sided, you can only reject if the test statistic is too large (say). You will therefore reject if the statistic is so high that it is in the region of the expected 5% highest values. If you test two-sided, you will reject both if the tests statistic is too large and if it is too small. Now it should be clear that you cannot always reject the two-sided test if the one-sided tests rejects. Because if you did so, as the two-sided test is symmetric, you'd need to reject as often on the negative side, and that would give you 10% rejections, but you want to test at 5% level, so that'd be too much.
In fact, in order to reach a rejection probability of 5% for the two-sided test, you can only afford to reject half the time on the positive side than the one-sided test would reject. The two-sided test has more options to reject (namely on both sides), so it needs to reject less often on either side in order to achieve the same overall level of 5%.
"Translated" into p-values this means that the p-value, if observing on the right side, should be smaller than 5% only half the time for the two-sided test than for the one-sided test. This is achieved by multiplying the one-sided p-value by 2.
