Final answer:
Microtubules are responsible for moving organelles within a cell and are part of the cytoskeleton which also maintains cell shape.
Step-by-step explanation:
The structure involved with the movement of organelles within a cell is microtubules. Microtubules are a component of the cell's cytoskeleton and provide a network over which organelles can be transported. They are hollow structures composed of polymerized tubulin dimers and along with motor proteins, they carry organelles across the cell. This facilitates various cellular processes, including intracellular transport and the maintenance of cell shape.