Yes, it goes on your record.
As for the job applications, no contest will show on a background check as a guilty plea, so if you leave it off, the employer will probably think you are dishonest if they do a check.
Good luck with your situation.
Under Penal Code §1203.4, a person 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 detention, 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.