Logistic regression

I'm studying for my comprehensive exams. I have read several texts on logistic regression and have found discrepancies between the definition and formulas for odds and odds ratio. Can anyone help me to understand the differences between the odds and odds ratio and how they relate to logistic regression. In class, I was informed that the odds ratio is equal to the probability of being in a group divide by the probability of not being in that group. However, I have read a book that indicates that the odds ratio is not a probability but is e^(B1)/ 1+e^(B1). Can anyone help me clear this up???:confused: