Diffusion is a passive process involving the movement of molecules from a region of a higher concentration to one of a lower concentration (the term passive means that the process does not require an input of energy to take place).
Diffusion can occur across partialy permeable membranes, such as those surrounding cells. Therefore, diffusion is involved in the movement of important molecules into and out of cells. It is important for the uptake of substances needed by cells, and also the removal of waste products produced by the cells.
In animals:
Respiration - Oxygen and glucose react to form carbon dioxide and water along with ATP (a source of energy) in the process of aerobic respiration. Therefore, oxygen and glucose must be taken up by the cell, and typically the concentration of these molecules outside the cell is greater than inside. Therefore, the overall net movement of these molecules will be down the concentration gradient, and they will move into the cell via diffusion. Similarly, the carbon dioxide produced is a waste product and moves out of the cell, again via diffusion down its concentration gradient.
In Plants:
Mineral uptake - Useful minerals and ions need to be taken up from soil into plants via root hair cells. These cells are adapted (through a large surface area and large number) to maximise the rate of diffusion. Therefore, the useful molecules in the soil move down a concentration gradient and into the roots to be taken up by the plant. Many molecules found in the soil are essential for the growth and survival of plants, making diffusion a very important process.