Two-tailed is for more exploratory research, or newer work, where you might not be sure of the direction of the relationship between the variables (e.g. targeting a different population, such as: an effect among men might play out differently among women and if the research has not been done with women, then a two-tailed test is more appropriate).  One-tailed tests are more permissible when you have a very strong theory for why the relationship would go in a certain direction and there has been previous research confirming that direction in correlation using those same variables/measures/constructs. Hope that helps!

One-tailed is used when you already determine that one is better, or worse, than the other whereas two-tailed is used when you hypothesize there is difference between two groups  but do not know which one will be better, or worse. It is easier to get p

To more directly answer your question: the advantage (I wouldn't say advantage, more like a difference) with a one-tail test is that you get to put all of your "wiggle room" (the confidence interval) between your alpha level (e.g. .05) and 0 all at one end of the distribution. Two-tailed tests split the wiggle room (split confidence interval), with half (2.5) on one end, half on the other. So, a comparison between two distributions may reach statistical significance on a one-tailed tests, but perhaps not a two-tailed test.  It's not about advantages, it's about what is most appropriate given the state of the research (see my first answer :).  I'm sure others can contribute here. This is a complicated issue. 

Tests are carried out to control error-rates. Sensible limits for these error-rates are determined by the expected wins and losses for acting correctly or incorrectly, based on the test statistic that is calculated from the available data. To value wins and losses you need to give a specific alternative hypothesis (HA) that is tested, that is, you must specify some minimum relevant effect size. The null hypothesis (H0) is a rather technical requirement for the maths to work, but the main point of the whole testing procedure is the alternative HA. Only when this is specified does a formal hypothesis test make sense at all.*

The test is a formal strategy to make a decision based on some given data that limits the maximum expected loss associated with possibly wrong decisions. The losses are found by evaluating HA. Recall that "HA is true" is a short term for "the effect (difference between groups, treatment effect, whatever your research is about) is at least minimaly relevant". The size of this "minimally relevant effect" will define what you expect to loose if you wrongly fail to recognize such an effect, and also what you expect to loose when you wrongly assume an effect of at least this size.

The test tells you finally to either "act as if HA is true" or to "act as if H0 is true". Both actions are associeted with some loss if they were wrong. The aim is to limit or balance these losses, and this is technically achieved by wisely chosing desired maximum error-rates for these cases and to design the experiment to guarantuee that the expected error-rates won't be larger. This all critically depends on your HA.

Given the dichotomy of possible rights and wrong decisions, there are two different "kinds of errors" one needs to control:

falsely acting as if HA is true: type-I error, rate is controlled by alpha

falsely acting as if H0 is true: type-II error, rate is controlled by beta

Technically, the chosen values for alpha beta will define the sample size, then the test statistic is calculated from the data thas has been obtained from the experiment, and the chosen value for alpha will determine a "rejection region for H0" for the statistic. The test is then so see whether or not the statistic falls into this rejection region. If is is inside, H0 is not rejected (->"act as if H0 was true"), if it is outside, H0 is rejected (->"act as if HA is true").

Under H0 the statistic has a known expected value (for the t-test this is 0, for the F-test this is 1). The rejection region must obviousely be somwhere else. This region can be at larger values, it can be at lower values, and it can be at larger and lower values at the same time. This makes the test "one-sided" (the rejection region is only at one side of the expected value) or "two-sided" (the rejection region is at both sides of the expected value).

What to chose depends on how you would act in which case:

Would you act as if HA is true

a) only when the statistic is higher (lower) than expected under H0?

b) when the statistic is either higher or lower than expected under H0?

In case a) you can commit a type-I error only when the statistic is accidentally too large (too low), but never when it is accidentally too low (to large).

In case b) you can commit a type-I error in any case where the statistic is accidentally either too high or too low.

So these are two different decision strategies, and hence this impacts the expected losses and the selection of reasonable values of alpha and beta.

Practically the selection of a one-sided or two-sided test depends on the conviction of your colleagues/reviewers that you really follow the underlying strategy. Example:

A drug is designed to lower the blood pressure (BP). You run the experiment. You find a decrease. You report a that a one-sided test lead to the rejection of H0 so that one can act as if HA was true (the BP is decreased by some relevant amount) so that the expected loss on such a decision is balanceds as described in the methods section (but hey, who does this practically? Noone! So the whole tning is a farce anyway).

The cruial question is if the colleagues/reviewers do belief that you would have ignored to report an increased blood pressure as "significant", that is you would not have rejected H0 for whatever observed increase. If this question can be answered with "yes" then a one-sided test was in place. Otherwise the reviewers belief that your decision strategy and your effective regection region was "two-sided", so reporting a "one-sided test" is actually cheating.

On the other side, for a well-planned RCT that is a priori registered there should not be such a problem. The registration documents should document the decision strategy, and if this is one-sided, then the study is designed to hold alpha and beta for a one-sided test, and so a one-sided test must be conducted, and an effect should only be reported as "significant" when the direction is as assumed. There cannot be any cheating (unless the reviewers miss to read the registration documents of the RCT!)

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*Do not confuse this with significance tests that just calculate a p-value under H0. Such tests are not a formal procedure and p-values alone are no sensible decision criterion; it needs a consideration of the observed effect, the sample size, the aim of the experiment, and much more to make a decision, and after thinking about all this it turns out that the p-value does not even add some useful information at all. - Nevertheless p-values are a kind of "normalized noise-to-signal-ratios" or "surprise indices" that may be used informally to rank a set of results. Further, sometimes interpretable effect sizes are not available ans so p-values are a valuable tool to get an impression of how much one should be "surprised" of such observations when H0 was "true". The often spread conglomerate of hypothesis testing (controlling error rates in a resonable decision strategy given a relevant HA) and significance testing (no specification of a relevant HA, only calculate a p-value under H0) to NHST (null-hypothesis-significance-testing) where only H0 is specified but is rejected at an a-priory defined, fixed level (often p

Thanks to you all for your invaluable contributions.