A photo is... a photo (a photograph). A picture can be a photo, a painting, or it can have an abstract meaning (for instance: "don't lose sight of the big picture!", that is to say "don't just focus on details, consider what you're dealing with, or studying, etc, in its wholeness, in its real meaning, etc"). "The pictures" (plural) can mean "the cinema".

A photo is produced by a camera or similar device. A picture can be a photo, a drawing, or a painting.

You didn't ask about painting, but other people commented on it. In the US painting is, yes, a representation that is drawn by hand (usually with paint, as opposed to a "sketch", which could be by pencil or something).

I agree with the other answers but would add that "picture" is very commonly used in the US to mean photograph. "He took my picture" is something you would hear all the time over here. You could say "he took my photo" and that's fine too, but that's just because picture and photograph are used interchangeably in the US and that's perfectly ok. I can't speak to its use in the U.K. or other English speaking countries.

A "photo" is short for "photograph." A photograph is a kind of picture. A painting is a different kind of picture. A drawing is a third kind of picture.