An EEOC charge is a formal complaint filed with the U.S. Equal Employment Opportunity Commission (EEOC) alleging that an employer discriminated against you in violation of federal employment-discrimination law.
If you believe that you were laid off because of your race or national origin:
You file a charge with the EEOC describing what happened.
The EEOC can notify the employer and investigate.
The EEOC may request documents, interview people, and evaluate evidence.
The EEOC can potentially pursue the matter itself, particularly if it sees evidence of broader/systemic discrimination.
Eventually, the EEOC may issue a Notice of Right to Sue, which generally allows you to pursue the matter in federal court.
Importantly, you don't need a smoking-g-n statement like “we fired him because he's American.” The EEOC can consider patterns, statistics, comparator employees, hiring/termination records, and other circumstantial evidence.
Why this matters
Suppose 30 Black, White, Spanish, Chinese, etc. employees are laid off over several years, and many are replaced by Indians who subsequently hire predominantly Indian employees.
An individual employee might file an EEOC charge saying, essentially:
“I believe I was selected for termination because of my race/national origin. I also believe my termination was part of a broader practice in which similarly situated American employees are being replaced by Indian employees.”
The EEOC could potentially investigate the broader pattern, rather than simply asking whether that one employee was treated improperly.
That's one reason an EEOC charge can be significant even if a severance agreement contains a release. A severance agreement generally cannot prevent someone from filing a charge with the EEOC or participating in an EEOC investigation.
However, you generally can't wait indefinitely. For most private-sector Title VII discrimination claims, the deadline is 180 days, extended to 300 days in many circumstances where a state or local fair-employment agency also has jurisdiction.
So if someone was recently laid off, I'd speak with an employment attorney promptly rather than waiting to see how the company's hiring pattern develops.