For you to count on a fair legal system, judges, jurors and lawyers must be able to handle their duties without undue bias. If any of these participants have a personal stake in your case that could call their objectivity into question, their service might be an unethical conflict of interest.
If you discover that someone presiding over your case had a conflict of interest, you might be able to use that fact as justification for an appeal after conviction.
Demonstrating unfair circumstances
Your lawyer has a duty to act in your interests. If your attorney had a close relationship with someone who could benefit from you losing the case, you might not receive the standard of representation you deserve.
Once your attorney uncovers a potential conflict of interest, the individual or the firm must determine if a conflict actually exists and if you may consent to further representation. To consent, you must provide informed approval in writing. You may have grounds for an appeal later if you demonstrate that your counsel discovered a conflict of interest and somehow mishandled the matter in a way that harmed your case.
Determining a conflict of interest by other parties
Jury selection is extensive to ensure jurors and alternates have no biases or personal connections to your case or similar cases. If a juror failed to disclose such a bias, you might have grounds for appeal. Likewise, if you can prove that a judge or juror could benefit financially or personally from your conviction, you might be able to establish a conflict of interest.
A conflict of interest can be a regrettable miscarriage of justice. Though the misconduct might be apparent to you, proving that fact to the court may take a lot of work.