There are several concepts which are linked together: AML (anti-money laundering), CTF (counter terrorism financing), KYC (know you customer), and PEP (politically exposed person).
The goal is to make sure the account is not used to manipulate funds of dubious origin or with a dubious intent:
- proceeds of crime
- financing of crime or terrorism
- corruption
- tax evasion
...
It all starts with KYC: know your customer. That's why when you open an account you need to prove your identity, tell them what you do, how much you make and so on. The bank (or other financial or payment institution) will check your identity against some databases, including list of politically exposed persons (PEPs), which are people in a position of power (ministers, MPs, top managers of large public companies, etc.), which could be subject to corruption, and more. PEPs will be subject to more stringent checks, and of course anyone on sanctions lists, linked to crime, terrorism, or countries under sanctions will usually not be able to open an account, or be under careful observation.
Once they know who you are and open an account for you, checks are made on incoming and outgoing operations. Anything above a certain amount will be reported to the relevant authorities (e.g. Fincen in the US, Tracfin in France, etc.). Anything out of the ordinary likewise. If you send or receive money to/from a suspect person, organisation or country, you will be flagged. If you suddenly send or receive large amounts of money which don't match you usual income or spending, you will be flagged.
In some cases they will ask you questions. It might be a perfectly legitimate operation that's just a bit out of the ordinary. In other cases they will just report to the authorities and it's up to them to follow up, possibly cross-referencing with other reports or any other information they may have. Depending on the nature of the unusual or suspect transaction, it may then be reported to the tax authorities (e.g. IRS in the US) if tax evasion is suspected, to the police/justice if other forms of fraud are suspected (money laundering, proceeds or crime...), or even to counter-terrorism (FBI in the US) if it may involve terrorism financing.
Note that even though in many places there are thresholds for operations to be reported (like 10K), splitting a transaction in a several smaller transactions (called "structuring") will not prevent the reporting (quite the contrary, it makes things even more suspect).
Sometimes all these measures mean that even the most innocent of people will be burdened with questions and delays and whatnot just because something unusual happens. But all these measures are not going away anytime soon (and banks which have not complied with those and turned a blind eye have regularly made headlines, getting huge fines, to deter others).