There is a direct relationship between the two.  The large the sample size, the greater the power to detect a true difference.  When setting up a study, I have first determined from the research design the power needed in order to reject the null hypothesis.  This will then determine the sample size for the study.  Most studies will traditionally allow for a 5% error when committing a "type 1" (alpha) error where the null hypothesis has been falsely rejected thus accepting the alternative hypothesis.  On the other hand, the traditionally acceptable error of 20% is acceptable to commit a "type 2" (beta) error where the null hypothesis is falsely accepted.  The power of the test (1-beta) determines the probability to correctly reject the null hypothesis and an increasing sample size will minimize both alpha and beta.  Hope that helps.

As Wayne indicates ... as sample size increases power increases for null hypothesis significance testing methods. This happens because the larger sample size reduces the standard error and increases the degrees of freedom. Remember that there are other factors that also affect power such as alpha level, 1 tail vs 2 tail test , and effect size.

Hope this helps.