england-and-wales
In England and Wales, with the exception of private prosecutions, complainants don't 'press' (or drop) charges.
There are several public authorities that can prosecute criminal offences but generally we talk about circumstances that involve the police and the Crown Prosecution Service (CPS).
When a crime is reported to the police the police investigate. When the police complete their investigation they refer the case and send the evidence to the CPS. (A minor offence such as low value shoplifting can be handled by the police, although if the case goes to court it must be reviewed by the CPS before the first hearing.)
Generally the CPS decides whether to prosecute based on 'the Full Code Test'. In short:
"Is there enough evidence against the suspect to provide a realistic prospect of conviction?"
"Is it in the public interest to prosecute?"
(In urgent situations the CPS might decide it's necessary to decide based on 'the Threshold Test'.)
The CPS prosecutes on behalf of the Crown, not the complainant. A criminal case is named along the lines of Rex/Regina or R. v Alex, not Bob v Alex.
Sometimes it may be difficult to proceed with prosecution if a complainant (or witness) decides not to cooperate or sometimes the CPS might decide not to prosecute because of the complainant's circumstances. But the CPS is allowed to prosecute without the complainant's approval or cooperation.
Under the Victims' Right to Review Scheme, in some circumstances a complainant can seek a review of a CPS decision not to prosecute or decision to stop a prosecution.