If you are modeling binomial data; ie a numerator consisting of the number of 1/0 successes you have for a given pattern of covariates, and a denominator that gives the value of the total number of observations having that covariate pattern (a specific profile of predictor values; eg age 23, married 1, working 0), a logistic regresson is generally appropriate. But when the mean values of the numerators are less than 10% of the mean values of the denominator, it is likely that a Poisson model is preferred. The otherwise logistic numerator is the count response variable (dependent variable) and the natural log of the denominator is the offset. Generally the Poisson model will fit the data better. Logistic models are not indended for rare occurrences.