It refers to the English legal system. In some countries if you break the law, you automatically get taken to court - 'prosecuted'.
But in England there are lots of conflicting systems. The main two are the civil and the criminal law (that's a gross over-simplification, but it'll do ). If you commit a 'civil' offence - that is, broadly, one that doesn't interest the police - you can be taken to court - 'sued' - by another person. If you commit a crime, you're not always sued (although you may be if the victim thinks the police haven't taken the matter seriously enough).
Two examples: person A is robbed. The police arrest the thief - a homeless man. But person A takes pity on him (the man only stole the price of a pint of beer) and decides not to 'press charges' - the police don't prosecute.
Person B is stabbed. He knows who did it, but the police can't prove it, and after a few weeks they give up trying - again, they don't prosecute. But Person B is not satisfied and employs a private investigator to find evidence of the stabbing. The standard of proof in the two legal systems is different*. He takes the aggressor to court, for medical costs, legal fees, and loss of earnings (this is not prosecution, because it's not a criminal court) and wins. The person who stabbed him doesn't go to prison, but he pays a civil penalty (often just money - but sometimes a large amount).
Sorry this is so long. As Pascal once wrote - I apologize for the length of this letter. I didn't have time to make it shorter [maybe I've quoted that before - it's one of my favourites].
criminal court: 'innocent until proved guilty'... 'beyond a shadow of doubt'
civil court: 'balance of probabilities'
PS You may see signs in the UK saying 'KEEP OUT. Trespassers will be prosecuted'. People who put up such signs don't understand the law. A trespasser (civil miscreant) can't be prosecuted. Some sorts of trespass are defined as 'criminal trespass', but the land-owner doesn't have the power to prosecute. All he can do is press charges, which can't arise, since - by definition- criminal trespass is something that the police will prosecute over without asking the land-owner's opinion (although he may be 'called as a prosecution witness'). So what these signs mean is
Trespassers, if arrested for criminal trespass, will be prosecuted
Trespassers, if caught, will be sued
Not so pithy