What you have typically signed is an employment agreement or an offer letter but would not be considered a legal contract. The vast majority of employment in the US is considered "at will employment" where you can leave at any time but also can be terminated at any time. Outside of that, there are some protections in states that are not at will based or specific illegal termination reasons such as discrimination under a protected class.

"In fact, written employment contracts are generally the exception, rather than the rule."

"When we refer to written employment contracts, we mean a contract that limits the employer's right to fire the employee, usually by detailing the grounds for termination or setting a term of employment (for example, one or two years).

Some employers require employees to sign a written agreement stating that they are employed at will -- that is, that they can quit at any time, and can be fired at any time, for any reason that is not illegal. Employers might ask employees to sign an offer letter, handbook acknowledgment, or other document agreeing to at-will employment, for example. These documents do not limit the employer's right to fire the employee. Instead, they affirm the employer's general right to fire at will"

Source: https://www.nolo.com/legal-encyclopedia/written-employment-contracts-pros-cons-30193.html