In the Neo-Assyrian Empire, provinces were organised into two types which affected the 'tax' you would pay. Land of Assur and Yoke of Assur.
Land of Assur was predominantly what happened to the territory of conquered states. The land was added to the domain of 'Assur' which was both the national god as well as the capital city so it has a double-meaing. A Land of Assur was ruled as a province with a governor (usually a member of the ruling family or similar), the native regime was removed and it was integrated into the empire.
The Yoke of Assur was basically a client-kingdom. They were fully autonomous with the native regime still in place, although it is most likely they acted as political puppets. They maintained their own army too and were expected to provide military support or to act on behalf of the Assyrian King.
The type of tax depended on which type. Those under a Land of Assur were expected to produce annual gifts to the temples which usually came under foodstuffs. There was also a 'microtax' if you want to call it which the provincial governor would levy upon the inhabitants in order to provide the annual gifts and more. This type of tax was on land produce such as grain and livestock.
Those under a Yoke of Assur were expected to give tribute. Such tributes were given directly to the king and were items such as precious metals, animals or other specialities. Failure to deliver on the agreed tribute would be seen as a political statement.
In terms of how it was collected. For Land of Assur, provincinal governors were responsible for the collecting of tax and used that tax to then provide the gifts for the temples in Assur (city) as well as to pay soldiers and such. Such a tax may not have even passed through the Imperial Palace. For the Yoke of Assur, tributes would have been delivered directly to the palace of the king.
There is also evidence of smaller taxes being levied such as at ports which would have been collected by port officials.
Sources:
Postgate, J.N. (1992), 'The Land of Assur and the Yoke of Assur', World Archaeology Vol. 23 No. 3: 247-263
Sharlach, T. M. (2004), Provincial Taxation and the Ur III State, Leiden/Boston.