Both are parametric test.
The t-test uses a test statistic that is related to the sample mean(s) and is used to compare that with the mean of another sample or some population.
The F-test uses a test statistic that is related to the sample variance and is used to compare that with the variance of another sample or some population.
Both tests require identical independently distributed random variables. This ensures that the relevant test statistics are approximately normally distributed.
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It appears that you might be referring to a situation in which Welch's t-test can be applied. Since it would be excruciating to write the required formulae here on answers.com let me refer you to the wikipedia page.
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P | T T F F Q | T F T F Q' | F T F T P + Q' | F T F F The layout is the best I could do with this software. Hope it is OK.
The answer depends on what is being tested: the t-test, F-test, Chi-square, Z-test are all commonly used with the Normal distribution. There are many others.
f=t/t; t=1/f