The everyday meaning of saturated usually refers to a sponge completely soaked in water. In chemistry, the concept applies to liquid solutions and molecular bonds. When you add a solid powder to a liquid solvent, the powder dissolves and disappears.
Eventually, the liquid reaches a point where it cannot hold any more of that solid. This specific state is called a fully saturated solution. If you add more solid to a saturated solution, it simply sinks to the bottom.
The extra material will stay completely undissolved in the container. At this saturation point, the mixture sits in a state of dynamic equilibrium. This means new solid dissolves into the liquid at the exact same rate.
Meanwhile, dissolved particles turn back into a solid crystal at that identical speed. Because these two rates match perfectly, the overall amount of dissolved solid never changes. Temperature plays a massive role in finding this exact limit.
Heating a liquid solvent usually allows it to dissolve much more solid material. In organic chemistry, the term means something entirely different. A saturated hydrocarbon is a molecule containing only single covalent bonds between its carbon atoms.
Because there are no double or triple bonds, the carbon chain holds the maximum possible hydrogens. This is exactly why solid animal fats in our diet are called saturated fats.