The IRD explains it nicely here. As they put it:
The Foreign Account Tax Compliance Act (FATCA) is United States of America (USA) legislation that aims to reduce tax evasion by USA citizens, tax residents, and entities.
Essentially, it's a reporting regime to make sure that USA persons (and New Zealanders with accounts in the USA) meet their tax obligations.
As part of FATCA, USA citizens/tax residents who have certain foreign financial assets that exceed certain thresholds must report those assets to the United States Internal Revenue Service (IRS), whether they live in the USA or not.
Who does FATCA apply to?
FATCA requires foreign financial institutions (namely, financial institutions outside the USA) that are not exempted, to register and report to the IRS on details of financial accounts held by USA citizens, tax residents and others.
If you are a US citizen or your parents are:
Please note that if you fall into one or both of these categories, then you will now be required to file yearly tax returns with the USA, even if you have never lived or worked there. There are firms that specialise in this; please also see this article.
I'm not a US citizen or tax resident. Why is my bank writing to me about it?
You'd have to ask them that. Probably covering themselves to make sure they are compliant with FATCA.