Literally speaking, a "record" just means any document, physical or electronic, that can be accessed in order to obtain information. Using this rather expansive definition, there are multiple places where "criminal records" reside. The very first place that has a record of all arrests, and copies of all police reports, is the police department itself. The second place is the local prosecutor's office. Finally, the courthouse that actually handled the matter will have "complete" (though often rather difficult to find) records of everything that took place in every case (usually referred to as the "minute orders" of the case).
Though all of the above "records" exist with information about a person's arrests and convictions, it is extremely unlikely that anyone who wants information about a specific person will ever see them. That is because the records are spread out and the person looking for the information wouldn't usually know whom to ask. Virtually every city in the nation (and there are a lot of cities in this great nation) has its own police force, and at least one courthouse. As for the prosecutor's office? The records they keep are not made public. Adding to the trouble is that each police department and each courthouse charges anyone who wants to search through and copy its records.
Historically this was, in fact, a problem. The California Department of Justice stepped in to solve this problem.
The California Department of Justice keeps a file on everyone who is convicted of a crime, or even arrested for a crime, within the state. These files are commonly referred to as a person's "criminal record." On the street (or in movies supposedly taking place on the street) you will sometimes hear a person's criminal record referred to as his or her "rap sheet." 
Who Can See My Criminal Record?
Anyone who wants to see it. Almost all employers now run criminal background checks. So do many lenders such as banks and credit card companies. If you are looking to buy a car you need to finance, a house you will need a mortgage on, or you need to get a job to pay for those things in the first place, someone is going to a run a criminal background check on you.
How Can Someone See My Criminal Record?
Very few people are going to go police station or courthouse hopping. It is far too time-consuming and expensive to do so.
Most people who want to do a criminal background check will pay a private agency to run a search through the records of the California Department of Justice and report back to them. The criminal records housed in the California Department of Justice are public records (meaning they are open to be viewed by any member of the public). There are dozens (possible hundreds) of private companies that will obtain these records for a price and give them to customers.
How Important Is This To Me?
Very Important. The people running a criminal background check on you are doing it for a reason. Those kinds of checks provide instant information about the prospective job/loan/rental candidate. What do they say?
- Have a conviction for theft?
You can't be trusted not to steal from the company.
- A conviction for DUI?
You have impulse control issues and are unreliable.
- Drug Possession?
You're an addict.
- Drunk in Public?
You're a drunk.
- An arrest for ANY sex-related or violent crime?
Say goodbye to any job dealing with the public.
That doesn't apply to you? Maybe not, but most people aren't going to take that chance. Is that unfair? You bet it is. You committed one verifiable mistake and because it is a matter of public record (making it easy to find out) it ruins your reputation.
 The term dates back to Medieval England where to "take the rap" meant literally to be beaten (rapped) as punishment for a crime. A "rap sheet" was a list of all the times a person had been punished in this way. http://www.wordorigins.org/index.php/site/comments/rap_sheet/