Cases are dismissed only after charges have been filed. "No charges filed" is self-explanatory. If no charges were actually filed in your case, you should let the immigration officer know this. In some situations, "no charges filed" is looked upon more favorably than "case dismissed".
No charges filed means you were arrested or cited but the DA never filed the case. Charges dismissed means the DA did file the case but then later decided to dismiss it. The two most common reasons DAs dismiss cases are either for insufficient evidence or in the interests of justice.
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Case Dismissed = USually means there was a filing but was dismissed after filing.
Case Dropped = Can either mean prior to filing or post filing
No Charges Filed = There was never any charge filed with the court.
It is generally better to have "no charges filed" than your case dismissed. A case gets filed if the prosecutor has enough evidence against you that they think they can prove guilt. A dismissal after a filing can be for any number of reasons. But if the case never gets filed in the first place, that means there was no case (i.e., no crime was committed or there is a legal excuse or defense), or that the case was weak, or the evidence insufficient, etc.
If your case was not filed, however, it does not necessarily mean it won't be filed in the future (so long as the statute of limitations does not expire). If the case was dismissed, on the other hand, the prosecutor cannot refile the charges if it was a misdemeanor. Felonies can generally be refiled once after a dismissal.
Case "dropped" could mean the case was not filed or it can mean it was dismissed.
Debra S. White, Esq.
Los Angeles Criminal Lawyer | Criminal Defense
Van Nuys Criminal Lawyer | Charges not Filed
This answer is based on California law and procedure and does not create an attorney-client relationship.