No federal laws that require employers to provide sick days to employees. However, in recent years, an increasing number of states and cities have passed paid sick leave laws to protect employees who are ill or looking after family members who are ill.
Currently, 15 states and the District of Columbia offer some sort of paid sick leave:
A handful of cities have passed similar laws, including San Francisco, Seattle, and Portland. If you're fortunate enough to live in one of these areas, you may be entitled to paid time off and protection from being fired.
If you don't live in a state with a paid sick leave law, you might have time off (albeit unpaid) under federal law. However, these laws usually apply only in cases of serious illness or disability.
For example, the Family and Medical Leave Act provides unpaid leave to employees with serious health conditions. The Americans with Disabilities Act provides reasonable accommodation to employees with disabilities, which may include unpaid time off.
Because it's common for employees to get the occasional cold or sickness, many employers choose to create sick leave or PTO policies, even if not required to by law. Check your employee handbook to see if your employer has a sick leave policy; if so, your employer must follow it.