What is the movement of water through the cell membrane in order to maintain homeostasis?

Osmosis. Cell membranes are "selectively permeable", where water is able to cross the cell membrane, but things that are dissolved in the water (solutes) are not. There are also small proteins called aquaporins which are designed to channel the movement of water across a cell membrane. You have to remember that water will move to achieve an equal solute concentration with that exists outside of the cell. A Hypotonic solution means that the liquid outside of the cells has LESS solutes in it than the cell does. Cells will take up water to match the low solute concentration outside the cells. A Hypertonic solution will have MORE solutes than the cell, and the cell will lose water to match the concentration outside the cell. Cells in a hypertonic solution will shrivel up, and cells will take up so much fluid in a hypotonic solution that they may burst.