Suppose someone in the United States has signed a contract promising to give his employer a certain amount of notice (say, one month) before terminating his employment. If the employee decides to ignore this, informs his employer one day that he quits and is not coming anymore, what are the employer's legal recourses?
Legally, relatively little. Most repercussions revolve around how much this hurts the former employee's reputation and future employability.
The employer may choose to claim for damages caused by the employee breaching their contract, however this is difficult. The employer would have to prove that damages were caused by the employee suddenly leaving, for example, having to employ a temporary staff member at short notice for a higher cost.