Yes - the arrest will show up on your criminal record. It will show the arrest and "no disposition."
The only way to get this off your record is if the court grants a petition for factual innocence. It's a fairly technical motion, but if granted, the court orders any arrest reports sealed and destroyed and the entry removed from your record.
It requires that the judge be convinced there was not probable cause to arrest you. The fact that charges were not filed certainly helps, but is not the bottom line. It also depends on how long ago this was. If it was a recent arrest, the case may still be under investigation, explaining why you were released without charges.
Mr. Dane is correct that the arrest will show up on your criminal history, commonly known as a "rap sheet." However, that isn't available to the general public.
When you say the you were "never charged and the case was dropped, " do you mean the District Attorney never filed charges? If the case was filed and dismissed, it will still show up on court records unless you get a factual finding of innocence, and Mr. Dane suggests.
If it was never filed, then there is no entry in the court's public records. 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.
HOWEVER, sometimes the DA won't file charges within the required two court days of an arrest, and the person is released... but charges are filed later. The District Attorney has a year to file charges on most misdemeanors, and three years or more on charges that can be filed as a felony.
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 this site, 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.