Final answer:
The centrosome is the major microtubule-organizing center in animal cells, containing a pair of centrioles that are cylinders of nine triplets of microtubules each.
Step-by-step explanation:
The major organization center for microtubules in all animal cells is the centrosome. This vital cellular structure is located near the nucleus and is integral to the cell's structural integrity, facilitating the assembly of microtubules. The centrosome consists of a pair of centrioles, each being a cylinder made up of nine triplets of microtubules, which are arranged perpendicularly to each other. The centrosome plays a key role in cell division as well as in facilitating the organization of the cell's internal framework.