Something that is functioning is working — doing what it's supposed to do. A functioning refrigerator keeps your food cold. A functioning television shows a clear picture. A functioning group gets things done in an orderly and timely fashion.
A function is the purpose that something is made for. A flashlight's function is to light up. But if it's not functioning, or working properly, you might need to change its batteries or buy a new one. If your car can get you from point A to point B but the engine sounds like it's going to die any minute, you might say that it is "barely functioning." Sometimes people use the word functional to mean the thing as functioning.