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Centrioles are cylindrical structures found in animal cells and some lower plant cells. They are composed of microtubules, which are protein filaments that provide structural support to the cell. Centrioles typically occur in pairs, perpendicular to each other, and are involved in cell division and organization of cellular structures. Now let's discuss their specific functions in nondividing and dividing cells.

In nondividing cells, centrioles play a critical role in organizing the cytoskeleton – the network of protein filaments that maintain cell shape, support internal cellular structures, and facilitate movement of organelles and vesicles within the cell. They serve as the microtubule-organizing center (MTOC), which is responsible for initiating the growth of microtubules. By controlling the growth and arrangement of microtubules, centrioles help to maintain the cell's structure and ensure proper cellular organization.

During cell division, centrioles play an essential role in the formation and function of the mitotic spindle, a structure composed of microtubules that helps separate chromosomes into two daughter cells. In animal cells, when the cell begins to divide, the centrioles replicate and move to opposite poles of the cell. The replicated centrioles then organize the assembly of the mitotic spindle by anchoring the spindle fibers to the centrosomes. These spindle fibers attach to chromosomes and ensure their even distribution to the two daughter cells during cell division. Once cell division is complete, the centrioles in the two daughter cells revert to their role as MTOCs, organizing the cellular structures and microtubules as required in nondividing cells. In summary, centrioles are essential for maintaining the structure and organization of both nondividing cells (as MTOCs) and dividing cells (by organizing the mitotic spindle and ensuring equal distribution of chromosomes during cell division).