Oral employment contracts (sometimes called "verbal" contracts) are simply contracts that are spoken and agreed to aloud rather than reduced to writing. It's common for employers and employees to enter into short oral contracts at the start of the employment relationship. For example, "if you can start tomorrow, the job is yours," followed by "I accept; see you tomorrow!" is a form of oral contract. If the employee quit his current job to take the new one offered in this conversation, he might have a claim for breach of contract.
Just because you can create an oral contract doesn't mean you should, however. Oral contracts are difficult to prove and enforce. If you and an employer are making a deal about something important, you should put it in writing.
In this country, most employees work at will: They can quit at any time for any reason, and they can be fired at any time, for any reason that's not illegal. (Illegal reasons for firing include discrimination and retaliation, for example.) If you have an employment contract promising you a job for a set term or stating that you can be fired only for good cause (or other reasons, such as defrauding the business), you are not employed at will. If the employer fires you during the contract term for a reason that isn't set out in the contract, you can sue for breach of contract.
An oral contract can change your at-will status. For example, if your manager tells you that the company doesn't fire anyone without good cause, or states that you will have a job at the company as long as you meet certain performance standards, that might change you from an at-will employee to one who can be fired only for those reasons.
It can be very difficult to prove the terms, or even the existence, of an oral contract. The problem may be a simple misunderstanding: Perhaps you and the person who hired you remember the conversation differently. Or, the problem could be intentional: Someone could lie in court about what was really said. Either way, it will come down to one person's word against the other. The judge or jury will ultimately have to decide who is telling the truth, and there's no guarantee that you'll win the fight. The better strategy is to always insist on getting agreements in writing. If there's a misunderstanding about the terms, you'll find out right away. And, if there's a dispute down the road, you will have solid proof of what everyone agreed to.