The Family and Medical Leave Act, or FMLA, is the federal law that requires covered employers -- those that employee 50 or more people within 75 miles -- to provide up to 12 weeks a year of unpaid leave to eligible employees. The leave could be for an employee's serious medical condition or for the serious medical condition of an employee's immediate family member.
The FMLA is a complicated law, though, with very specific requirements and stipulations. For example, the law specifies that only your spouse, children or parents are considered immediate family members. In other words, you wouldn't be covered by FMLA if you took off work to care for your brother or sister. That is, unless your brother or sister is your next of kin and a military service member.
Also, if you and your spouse work for the same company and you both intend to take unpaid leave under FMLA after your baby is born, then the maximum time you both can take off to care for and bond with your child is a combined 12 weeks. In other words, you have to split the 12 weeks between you if you take off together.
However, you are still entitled to the difference in time off if you had to take less time because you and your spouse were on leave together. That is, if you and your spouse took off for six weeks simultaneously, then you could still take off another six weeks in the same year if you need to tend to a serious health problem.
A helpful article in U.S. News & World Report touches on some other complicated aspects of the federal FMLA.
To learn more about FMLA eligibility and qualifying reasons for taking leave, please see Serrins Fisher LLP's FMLA overview.