If you don't know what happened in your case, how do you expect to get meaningful advice? This is important, and could affect the rest of your life, so you need to spend the time to get the details straight. You should talk to the lawyer who represented you, or the clerk's office at the courthouse where your case was heard may be able to help you.
The only way to remove an arrest from your record is to get a finding that you were factually innocent of the crime; a dismissal, or even a not guilty verdict, is not enough. It sounds like you admitted guilt, so this would probably not be an option for you.
Note that employers cannot inquire about arrests that did not result in a conviction.
A plea bargain is an arrangement where you enter a plea of guilty or no contest in exchange for an agreed sentence. If you entered into a plea bargain, then you were convicted. There are also diversion programs, where you can avoid a conviction if you stay out of trouble for a predetermined period of time.
How old were you when this happened? If you were a minor, you can get your juvenile record sealed.
Under Penal Code §1203.4, an adult who is granted probation can apply to have the case dismissed after probation is complete. This is sometimes called an expungement, but it doesn't really expunge anything or seal your record. The conviction is still a part of the court's public records, which will also show the case was dismissed. You still must disclose the conviction when applying to be a police officer or for certain other jobs, or if asked on an application for a state license or to run for public office. It will not remove any restrictions on your right to own a firearm or relieve you of the requirement to register as a sex offender. The conviction can still be considered a prior offense; for instance, a prior theft conviction could make any future petty theft a felony, or a prior DUI conviction could be used to increase the punishment in subsequent DUI convictions.
BEWARE of law firms that promise to "clear your record" and charge thousands of dollars. The process of filing a Penal Code §1203.4 petition is pretty simple, and you can do it yourself with forms available from the court clerk's office, especially if you completed probation without any problems. If you had a probation violation along the way, you may want an attorney's help, but the guys who charge thousands to file simple paperwork will probably overcharge you for that, too.
California Labor Code §432.7 says employers can't ask about any arrest that didn't result in a conviction, inquire about it from other sources or use it in a hiring decision.
Some attorneys interpret this Labor Code section to mean you don't have to disclose a conviction that was dismissed under Penal Code §1203.4, but I usually advise clients to disclose it, with an explanation that the conviction was subsequently dismissed.
Please understand that this is a general discussion of legal principles by a California lawyer and does not create an attorney/client relationship. It's impossible to give detailed, accurate advice based on a few sentences on a website (and you shouldn't provide too much specific information about your legal matter on a public forum like Avvo, anyway). You should always seek advice from an attorney licensed in your jurisdiction who can give you an informed opinion after reviewing all of the relevant information.
Mr. Marshall, as always, has given a detailed answer about the relevant code sections and procedures to seek an expungement.
The most important thing for you is to know exactly what you're dealing with. Can you handle expungements on your own? Yes, but if you aren't sure of what happened in your original case, I'd suggest contacting an attorney to unravel this for you and start you in the right direction to clear things up as much as the law allows.