First, employees are entitled to use any "sick time" that may be made available to them by the employer and according to the employers policies and procedures.
Regardless of the policies and procedures, some employees may be entitled to time off for their illness or to care for a family member pursuant to the Family and Medical Leave Act (FMLA). Generally, the FMLA provides that an employee may take up to 12 weeks of unpaid leave to care for the serious health condition of themselves or a family member. However, the FMLA only covers certain employers. To be a covered employer the employer must employ at least 50 employees. Additionally, an employee does not generally become eligable to receive FMLA time until they have been employed for 1 year.
Employers face many pitfalls in implementing proper FMLA policies and should seek competent counsel in doing so. Likewise, if you are an employee and feel that your rights to FMLA leave have been violated you should seek competent counsel.