Substances are transported passively down concentration gradients in diffusion and osmosis. Often, substances have to be moved from a low to a high concentration - against a concentration gradient.
Active transport is a process that is required to move molecules against a concentration gradient. The process requires energy.
For plants to take up mineral ions, ions are moved into root hairs, where they are in a higher concentration than in the dilute solutions in the soil. Active transport then occurs across the root so that the plant takes in the ions it needs from the soil around it.
In animals, glucose molecules have to be moved across the gut wall into the blood.