An employment contract is an agreement between the employer and the employee about the terms of employment. If you have an employment contract, and your employer breaks ("breaches," in legalese) it, you may be entitled to damages.
In the United States, employees are generally presumed to work at will, which means they can quit at any time for any reason, and they can be fired at any time, for any reason that's not illegal. (It would be illegal, for example, for an employer to fire even an at-will employee for discriminatory reasons.)
Some employment contracts don't change the at-will relationship. For instance, an employer might send an employee an offer letter, stating what the job will be, when it will start, and how much the employee will be paid, but also stating that the employee will work at will. This creates a sort of employment contract, but it doesn't limit the employer's or employee's right to end the relationship.
On the other hand, some employment contracts limit the circumstances in which an employee can be fired, or they impose penalties if the employer ends the contract early. For example, an executive might be hired for a two-year term, during which she can be fired only for gross misconduct. Or, a contract might provide that the employer can end the employment relationship on 90 days' notice or pay in lieu of notice.
A contract is breached, or broken, when either party doesn't live up to its agreement. For example, if you have an employment contract promising that you will be paid an annual salary of $50,000, but your employer decides to start you at a lower amount, that would be breach of contract. Similarly, if you have a two-year contract stating that you may be fired only for good cause during the contract term, and you are fired after a year and a half so the owner of the company can put his nephew in your job, that would be breach of contract.
If one party breaches a contract, the other party can sue for damages to compensate for the financial harm caused by the breach. In the employment context, this often boils down to salary, benefits, and other amounts an employee either doesn't receive or has to pay because the employer ended the employment relationship in violation of the contract's terms.
Example: Jerry is hired as CFO by FunCo. His employment contract states that he has been hired for a three-year term, during which he can be fired only for gross misconduct. If Jerry is fired after 30 months for any other reason, he would be entitled to sue for six months of salary (the time remaining on his contract), plus the cost of any benefits and other perks he has to pay for out of pocket.
An employee can't just sit around and collect a paycheck, however. The employee must try to find new work. This is called "mitigation of damages." If the employee finds a new job during the contract term, those earnings will be subtracted from what the employer owes for breach of contract.
Example: Back to Jerry, who is searching diligently for work. He can't find another CFO job or a position that pays as well, but after three months, he finds a job that pays him $5,000 a month. If Jerry was earning $8,000 a month as CFO, he would be entitled to $33,000 in damages for his lost salary: Three months of $8,000 per month while he was out of work, plus the difference between what he used to earn ($8,000) and what he earns now ($5,000) for the remaining three months on his contract.