The phrase "to press charges" means that a victim of a criminal action reports that action to the police, filing a police report so the district attorney or local prosecutor can then prosecute a case. Generally, this causes criminal charges to be brought by the prosecutor against an accused person. Once those criminal charges are filed, the accused may be arrested and will have to either arrange a plea bargain with the prosecutor or will have to stand trial for the crime and potentially face jail time or other penalties.
Pressing charges is different than suing in civil court. When a person decides to take this action, he will not benefit monetarily as a result of the outcome of the case. Instead, the case will essentially be out of his hands and handled by a prosecutor; he may be called upon to testify at a trial or to provide evidence to support the charges, but the onus is ultimately on the prosecutor to create a theory of the case, to collect evidence and to try the case. In a civil suit, on the other hand, the plaintiff sues and must prove that the defendant injured him either negligently or on purpose; the plaintiff then recovers monetary damages if he can prove these elements.
If a person decides to press charges, he must report the event that occurred in as much detail as possible. The prosecutor will then review the information provided and determine whether to prosecute or not. Not every situation leads to an arrest or trial. Sometimes, the prosecutor will decide there is insufficient evidence to arrest the accused and take him to trial; other times, the prosecutor will determine the behavior of the accused did not meet all the elements of the crime and therefore no criminal sanctions are appropriate.
At times, a prosecutor may also try a case even if the victim decides not to press charges. Because a law has been broken, the accused defendant's actions are not just a crime against the victim, but also against the state and its laws. As such, a prosecutor doesn't necessarily need a victim to cooperate, and he can subpoena or compel a victim to testify to prove his case if necessary. This may occur in domestic violence cases where a victim may be unlikely to wish to prosecute his or her her significant other, especially if he or she stays in the abusive relationship.