As with most statistics questions, it depends.  Traditionally, most would argue using a two-tailed test.  One-tailed tests are used very rarely.  It depends on your alternative hypothesis.  Assuming your null hypothesis is that there is no relationship between your variables (i.e., r = .0), then a two-tailed alternative hypothesis would be that there were some relationship, either positive or negative.  A one-tailed would alternative would specify the direction of the relationship (i.e., r is definitely positive or r is definitely negative).  It is more conservative to run a two-tailed test and also honors conditions in which the relationship in your data may come out in a way you did not expect.  You should have a very good idea the direction of the relationship before you even collect data but again, data doesn't always conform to our hypotheses, so it is probably best practice to run the more conservative two-tailed test. 

In using a one-tailed (directional) test you are assigning zero to the tail probability in one direction. Therefore in a strict sense it is only valid if there is zero chance you would conclude that there is an effect in that direction. So I would generally advise using such a directional test if there is no circumstances in which you would conclude that the effect is in that direction. Generally that's only likely to be true if the result is impossible, or perhaps would indicate that something has gone badly wrong with the design of the study.

For example, I've used a one-tailed test in a memory experiment as a test of baseline learning. In this case if they are below chance this indicates they haven't learned anything at baseline or my test is biased and subsequent data are probably meaningless.

In contrast if I'm testing a new drug I might expect it to make patients better, but I think I would still be interested if it made them worse (and there are real examples where this has happened).

Thanks!

It depends, if you have stated directional hypothesis, one-tailed test will be appropriate, if it is non-directional hypothesis, two-tailed test is ideal. The latter is moslty used.

You have been given good advice - but are you sure that correlation is what you want ?

CORrelation is about association - that is the extent that two variables go together - it essentially measures the scatter around the line and not the line - the relationship itself., ( and that could be determined by the variation in the predictor) .

For example I would not usually calculate the correlation between age and alcohol consumption as alcohol  does not determine age - there is a one way relation so that regression is the go to measure; and you are then  able to do all sort of diagnostics about the validity of the relationship.

I am just encouraging you to think about it.