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The question is vague. Regression can be a complex analysis, and which information is important depends greatly on what you are using the results for.
But very generally, if you are using regression as a hypothesis test, then the F (test statistic), r-square (effect size), and p (significance level), will be important. If you are using regression for predicting a value of Y based on X, then the slope of the regression line (b) and its intercept with the Y axis (a) are needed for the regression equation: Y = a + bX.
Computer programs such as SPSS also test the statistical significance of both the intercept and the slope by comparing them to zero, and they will report several other numbers related to these tests. However, this may or may not be information that the researcher is interested in. Again, it all depends on the situation.
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