Typically it works like this:
Owed in the state that the sale is made (in your case, New York). This is difficult in the internet age with businesses operated with no physical presence in a state. Technically sales (and local taxes) are due depending on where the item that was sold is delivered, so if you shipped an item made in Texas to New York, New York sales taxes would be due on that item. These taxes would be paid to the state of New York. Same for all 50 states. This is a huge burden on small businesses and is not usually enforced. Most small businesses only collect sales tax on items sold to the state that the business is operating from (in your example, you would only collect Texas state sales tax).
This is the corporate tax levied by the state that the business is operating in, in this case it would be Texas. Since Texas does not have a state income tax, there would be no income tax. However Texas does have a Franchise Tax that would be due (this is due from all business organized or operating in Texas)
The state that the business is registered in may have its own taxes which would be on the income of the business and would be specific to the state that the business is registered in. These can also be in the form of registration fees, annual fees, franchise fees, etc.