I owe a company a debt, with regular monthly payment. Do they have the right to take my payment and apply it to whichever month they wish, or is it up to me where the payment goes?

I had a problem with them and they said their payment system (NPS?) automatically applies the payment to a certain month, which I objected to.

It depends on the contract you have with them. If you're talking about credit cards, the cardmember agreement specifies how payments are applied. Mortgages are the same way, and you have to specify if you want a payment to apply to principal vs. your monthly payment. Basically you need to reach some sort of agreement with that company, there's no hard & fast rule about how it works.