"Pressing charges" has no legal meaning in California. When a possible crime takes place, the police investigate the incident. The victim may or may not give a statement to the police. Once the police conduct a preliminary investigation, perhaps they have been able to identify the perpetrator. The police send their investigation to the district attorney. The DA's office has a "filing deputy" who looks over the incoming investigation reports and decides whether or not to file criminal accusations against a person, and how to frame them legally.
The DA does not take his marching orders from the complaining witness. Many times the DA's office files charges which most people disagree with, and now and again the DA refuses to file charges which laypeople think should be filed.
Essentially, once the matter is in the hands of the police, the aggrieved person has virtually no control over how the process unfolds.
That is a complicated question. If the offense is a misdemeanor not committed in the presence of a police officer, then the victim of the crime must decide he wants the offender to be prosecuted. If the charge is a felony, the victim's decision to "press charges" is not required for the D.A. to file the case. However, in a felony charge, the victim's wishes are important in the D.A.'s decision to prosecute and how to resolve the case.
"Pressing charges" is often confused with simply talking to the police. Only a district attorney may file charges against another person. Someone who is not a district attorney cannot do so, however, that person can certainly speak to police in the course of their investigation. That same person may speak to a district attorney, but the district attorney has discretion whether to follow the suggestions or desires of a victim or witness.
The reason for this is because a district attorney is trained and educated to know what is a violation of the law and what evidence is necessary to prove such a violation to convict a person. Were all citizens empowered to file charges, it is not only likely that many charges would be found meritless, but citizens would use the judiciary for an ulterior purpose, such as to harass or spite a person, perhaps due to jealousy or to gain a civil advantage.
In my experience, I often hear folks seemingly boast that "I pressed charges" against another person. Their comment implies they believe they have the power to dismiss the charges on certain conditions when in fact only the prosecutor or judge has such an ability. What the person seems to suggest, but in an awkward way, is that their interaction with police or even a prosecutor provided important or determinative information that led to charges being filed. Such folks believe that if they were to recant their story, charges would be dropped.