Generally, internet retailers are not required to remit sales taxes unless they have a physical presence in the state. The states other than NY in which Amazon remits tax presumably have a physical Amazon presence, such as warehouses, etc.
Given the horrific budget situation that NY is in, they recently passed a law that says Amazon has physical presence in NY because some of its partners, which sell products on Amazon, have a presence.
This is a blatant money grab by NY and is being contested by Amazon in the courts. They hope to win the court case and claim refunds for the sales taxes remitted.
Technically, in all states with a sales tax, you are required to remit use tax if you bring in a product on which you have not paid sales tax. Generally this tax is self-reported on the state individual income tax. As may be guessed, levels of compliance with this requirement are low.
As a result, several states have talked about requesting sales records from large online retailers to try to catch people with large unreported use tax liabilities. This is a pretty low risk for people as the total amount of use tax per consumer is generally low. For example, purchasing a $1500 television from Amazon and having it shipped to a state with a 6% sales tax would give rise to a $90 use tax liability. It would cost a state most of that to track down the individual.