Short answer: In an external review, an independent organization (not your insurer) re-examines the denial. The reviewer can uphold or overturn it, and the insurer is required by law to accept the decision, so it is effectively final. If a federal process handles the review there is no charge; otherwise any fee is capped at $25.
After you exhaust the internal appeal, you can take a denial to external review, an evaluation by an independent third party. It applies to denials that involve medical judgment, decisions that a treatment is experimental or investigational, and rescissions (coverage cancelled over alleged misstatements on your application).
Your insurer is required by law to accept the external reviewer’s decision, so the company no longer has the final say, and the result is binding on the plan. Standard external reviews are decided within 45 days. If your plan uses the federal (HHS-administered) process there is no cost; if it uses a state process or a contracted review organization, any fee is capped at $25.