What exactly are AML and KYC checks?

This is a pretty broad question. AML, which stands for anti-money laundering, is part of the Compliance Program that any regulated entity needs to implement. A regulated entity would be licensed and governed by a financial regulator.

Part and parcel of any AML manual are to have a decent understanding of KYC, i.e. Know Your Customer. This implies that you have to have as part of your onboarding process a methodology to correctly identify and authenticate a customer (KYC usually implies an individual). For business, the correct terminology is KYB (Know Your Business).

Once a customer has been identified, usually by using a government-issued photo-ID, and using ancillary information unique to the customer, then the customer is now tracked for any financial activity that they perform. Key importance is now to see if any financial transactions related to this customer would violate or flag or come close to the rules and thresholds laid out for detecting money laundering behavior.

This is precisely what an AML & KYC check would do.