It depends. The federal Family and Medical Leave Act (FMLA) requires you to grant leave for your employees to care for family members with a serious health condition, and it defines family members as parents, spouses, and children. (Parents include those persons who took the place of a parent when the employee was a child; children include those children whom the employee cares for and supports.) The definition, however, does not include many people that most of us consider family members, including grandparents, aunts and uncles, in-laws, same-sex partners, or siblings. For more on the FMLA, see Nolo's article Providing Family and Medical Leave.
The military family leave provisions of the FMLA might allow time off, however. These provisions, added in 2008, allow eligible employees to take up to 26 weeks of leave in a single 12-month period to care for a family member who was seriously injured while serving in the Armed Forces. For purposes of this entitlement only, the definition of family member is much broader, and includes adult children, siblings, grandparents, and cousins. For more information on these rights, see Nolo's article Family and Medical Leave for Military Family Members.
If your state has a family and medical leave law, it might require leave to care for siblings. To find out about your state's law, contact your state labor department. For information on employee leave rights under state laws -- and what happens if both the FMLA and a state leave law applies -- see Nolo's article State Family and Medical Leave Laws.