As far as I am aware it isn't a specific thing, but a general catchall term for going through the process of filing a criminal complaint, and seeing it through to completion. Maybe there is better words for it but "pressing charges" is what they use on TV so it is top of mind.

In general I meant there is a difference between criminal and civil law, and suing generally refers to civil not criminal law.

It is a thing. In America pressing charges is when you accuse somebody of a crime and ask a prosecutor to bring criminal charges against them.

Prosecutors exclusively decide who is charged. No charges can be "pressed" by a victim.

Yes, in most cases it is the prosecutor's discretion whether to bring a case to a grand jury, but that isn't what pressing charges is. See Merriam Webster's definition[0].

[0] https://www.merriam-webster.com/dictionary/press%20charges