A growing thing (or person) is in the process of developing, often by getting bigger. You can argue for a second helping of cake by saying, "I'm a growing kid!"
Your puppy is a growing animal, changing and developing into an adult dog, and your cactus is a growing plant, even if the only evidence of that growth is one pink bloom each year. You can also use this adjective for things that increase, like your growing bank account or your growing interest in woodworking. As a noun, growing means the biological process of change and development itself: "Growing isn't always easy, but we all have to do it."