The basic rule in Texas is the "employment at will" doctrine: absent an express agreement to the contrary, either party in an employment relationship may end the relationship or change the terms and conditions of employment at any time for any reason, or even for no particular reason at all, with or without notice.
There are several exceptions, both statutory and court-made:
statutory exceptions
state and federal employment discrimination statutes: a discharge may not be based upon a person's race, color, religion, gender, age, national origin, disability, or citizenship, and many states add veteran status and sexual orientation to the list
protected activity (something the law entitles an employee to do without fear of retaliation)
bringing suspected wrongdoing to the attention of competent government authorities (state and federal whistleblowing statutes)
filing various types of claims (OSHA, federal wage and hour, workers' compensation, employment discrimination, etc.)
military duty
jury duty
voting
engaging in union activity
common law exceptions (i.e., exceptions found in court decisions)
public policy: it is illegal to discharge an employee for refusing to commit a criminal act
contractual - if a discharge would violate an express employment agreement, it would be a wrongful discharge; includes collective bargaining agreements
In a recent Texas case, a court ruled against a company that had failed to enforce its anti-nepotism policy for 17 years and then suddenly fired an employee who was known all that time to have violated the policy.
Remedies for wrongful discharge can include reinstatement, back and future pay, promotion, punitive damages, and an injunction against future illegal conduct. In addition to compensating the employee, the employer can also be made to pay attorney's fees, expert witness fees, and court costs.