A photograph is a picture that is made by a camera. You can shorten the word to photo which is less formal. A picture is another word for referring to a photograph, especially in newspaper, magazines, ... In those cases you can also say illustration or figure, especially when referring to a diagram.
This illustration shows the process in form of a diagram.
In figure one, you can see that the value changed over the years.
A image is a picture that you can see in a mirror, on a TV, ... Again, you can use picture in that context.
A picture can also refer to a painting or drawing.
Moreover, both image and picture can mean an impression or mental image that you have of someone.