What remains and how it affects him depends on a few things - first, if he was a minor (under 18) at the time, he can seal his juvenile records. If he was an adult, the terms of the deal matter.
If he got a diversion under PC 1000, once he completes, the case is dismissed and legally speaking it's as though he was never arrested.
If he took a conventional plea and was sentenced to probation, he can seek a dismissal after completing probation. This is commonly called an expungement (but it doesn't truly expunge the case - it adds a notation that the case was dismissed and it can't be used against him in most employment situations.
The above answer is for general information only and is based on the information you posted. Every case is fact dependent, so to get a thorough analysis of your situation, you will need to consult face to face with an attorney licensed to practice in the jurisdiction where the incident took place. Do not conclusively rely on any information posted online when deciding what to do about your case.
If he successfully completes probation he will be eligible for expungement under CA Penal Code section 1203.4 . It will show up as "dismissed" yet he will have to still disclose it on state license applications and others. Here's the pertinent part of the code;
1203.4. (a) (1) In any case in which a defendant has fulfilled the conditions of probation for the entire period of probation, or has been discharged prior to the termination of the period of probation, or in any other case in which a court, in its discretion and the interests of justice, determines that a defendant should be granted the relief available under this section, the defendant shall, at any time after the termination of the period of probation, if he or she is not then serving a sentence for any offense, on probation for any offense, or charged with the commission of any offense, be permitted by the court to withdraw his or her plea of guilty or plea of nolo contendere and enter a plea of not guilty; or, if he or she has been convicted after a plea of not guilty, the court shall set aside the verdict of guilty; and, in either case, the court shall thereupon dismiss the accusations or information against the defendant and except as noted below, he or she shall thereafter be released from all penalties and disabilities resulting from the offense of which he or she has been convicted, except as provided in Section 13555 of the Vehicle Code. The probationer shall be informed, in his or her probation papers, of this right and privilege and his or her right, if any, to petition for a certificate of rehabilitation and pardon. The probationer may make the application and change of plea in person or by attorney, or by the probation officer authorized in writing. However, in any subsequent prosecution of the defendant for any other offense, the prior conviction may be pleaded and proved and shall have the same effect as if probation had not been granted or the accusation or information dismissed. The order shall state, and the probationer shall be informed, that the order does not relieve him or her of the obligation to disclose the conviction in response to any direct question contained in any questionnaire or application for public office, for licensure by any state or local agency, or for contracting with the California State Lottery Commission.
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I agree with my colleagues and add that it really depends on what drug possession charge he plead to, whether he was a minor or adult, whether he was on diversion pursuant to PC 1000 or treatment pursuant to Prop. 36, or a felony conviction for drug possession. Assuming that your son competes all of the requirements and successfully completes his diversion or probation period, there are various options depending on what the "plea deal" was. This question is better suited for the attorney who represented your son.