Under California law, generally speaking, all employment is presumed to be terminable at-will, without any advance notice (absent certain exceptions). At-will employees may be terminated for any reason, so long as it is not illegal. This presumption is codified in California Labor Code section 2922.
Employees who work under an employment contract can only be terminated for reasons specified in the contract.
In California, the at-will presumption can be overcome by evidence that despite the absence of a specified term of employment, the parties agreed who the employer's power to terminate would be limited in some way.
People are commonly under the misimpression that they have a lot more workplace rights than they actually do. Most Californian employees, and most U.S. workers are "at will," without an employment contract or labor union that protects their employment.
In contrast, employers in France need good cause to fire their employees. "At will" means your employer can fire you without any notice at all, and for any reason or no reason. Similarly, you're free to quit at your will, which is probably no comfort to you right now.
They can, but don't have to offer you any severance pay.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.