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Semipermeable membranes are vital parts of biological systems - they're a big part of how the cells of plants and animals work.
A permeable membrane is a biological or synthetic material that has tiny holes in it, allowing small particles (including water molecules and ions) to move through it. A semipermeable membrane is a membrane that only allows certain types of particles to move through it under certain conditions.
Osmosis is where solvent molecules (usually water) move from one side of a cell membrane to the other. This happens because the concentration of a solute is higher on one side. When two nearby concentrations are different, materials will tend to spread out to equalize the concentration. That's why, for example, if we put a drop of food coloring in some water it will gradually spread through the whole water.
One side of the membrane is often kept at a lower concentration on purpose to force desirable molecules to keep moving into the cell. The cell removes the molecules as soon as they arrive to keep osmosis happening. At least, that's the simple explanation.