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The logistic function is $$f(x) = \frac{1}{1+e^{-x}},$$ which maps real numbers to $(0,1)$. One common use of the logistic function is logistic regression, which is a standard method of quantifying the effect of a set of predictors $\{X_1, ..., X_p\}$ on a binary outcome, $Y$. The model can be written as $$P(Y=1|X) = f(\beta_0 + \beta_1X_1 + ... + \beta_p X_p)$$ The logistic regression model has the nice property that the exponentiated regression coefficients can be interpreted as odds ratios associated with a one unit increase in the predictor.
A second use of the logistic function (but unrelated to logistic regression) is the logistic distribution, which has $f(x)$ as its quantile function.