Fit Y by X, more specifically Bivariate, is performing standard least squares regression. Regression does not assume that your data be normally distributed. But it does assume that the residuals (or errors) be normally distributed. This distinction is important because the raw data will have signals in it that could prevent it from being normally distributed. The X or explanatory variable should remove the effect of the signals. What is left (the errors or residuals) need to be normally distributed in order for all of the statistical inferences of the regression to be valid. That is why a regression analysis should always include an examination of the residuals as well. Bivariate will allow you to perform the residual analysis.
If the residuals are not normally distributed, there are a few options. One is to transform your response so that the residuals can be normally distributed. The Bivariate report offers several options to transform your response. They can be found under Fit Special from the red pop up menu.
Another option is to use Fit Model and specify a Generalized Linear Model. That will allow you to specify the distribution response. A generalized linear model is more than I can get into here, but might be worth you looking into for whatever your problem is.
Dan Obermiller