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Is a normally distributed variable needed to have a normally distributed sampling distribution.?
Yes, it is.
What does it mean when you say a variable is normally distributed?
It means that the probability distribution function of the variable is the Gaussian or normal distribution.
What do results tell you with regard to whether the data is normally distributed?
Your question may be a little overly complex. Usually you start with some degree of confidence that the variable(s) you are studying are normally distributed; you don't do an experiment and wait for the results to tell you. Your confidence that a variable is normally distributed helps you to determine whether or not your results are significantly different from what you would expect by chance. If I have missed the meaning of your question, my apologies.
Why is the central limit theorem an important idea for dealing with a population not normally distributed?
According to the Central Limit Theorem, even if a variable has an underlying distribution which is not Normal, the means of random samples from the population will be normally distributed with the population mean as its mean.
When doing a t-test what does normal distributed in population mean?
It means that the random variable of interest is Normally distributed and so the t-distribution is an appropriate distribution for the test rather than just an approximation.
