Final answer:
The cell membrane is the organelle in an animal cell that facilitates diffusion and osmosis, due to its selectively permeable phospholipid bilayer structure that allows water and certain molecules to move across it.
Step-by-step explanation:
The organelle in an animal cell that allows for diffusion and osmosis to take place is the cell membrane. This membrane is composed of a phospholipid bilayer with embedded proteins that create a selectively permeable barrier. It permits the passage of water and small molecules while restricting larger molecules and ions. The process of osmosis involves the movement of water across this semi-permeable membrane, from an area of higher water concentration to one of lower concentration. Conversely, diffusion allows for the movement of other substances along their concentration gradient across the cell membrane.
This essential function of the cell membrane is vital for maintaining cell volume, pH levels, and concentration of ions and molecules necessary for life processes. The semipermeable nature of the cell membrane is what enables osmosis, as well as facilitated diffusion, which is mediated by protein channels and carriers, and active transport mechanisms that require energy input to move substances against their concentration gradients.