To soften, in cooking, is to reduce the firmness of an ingredient with gentle heat, moisture, or time so it becomes more pliable, tender, or easier to work with.
The most common use is with solid fats: leaving butter or shortening at room temperature, or warming it briefly, makes it spreadable and easier to cream into batters and doughs. Vegetables, dried fruit, nuts, and hard cheese are softened with gentle heat (sautéing, steaming, simmering) so they're easier to chop, mash, or incorporate. Dried ingredients such as beans, lentils, and dried mushrooms are softened by soaking or rehydrating in water or broth before cooking.
Softening is distinct from cooking through: the goal is a workable texture, not full doneness, so it's usually a preparatory step rather than the final stage of a dish.