What exactly does a cinematographer do compared to the film's director?

I'll be as simple as I can, the cinematography is the person who makes the technical things like camera, lights and sound, translate well into an actual moving film. The director is basically what it says on the tin, the person who gives direction and instruction to not just the actors but also the other various departments. In a film there will be a lot of roles that will be very similar to the main Directors role, that is because they must all work very closely together to produce the film.

Well, you asked me to go easy, so I won't jump all over you. The director NEVER handles the camera. Never did, as far as I know. That's not his job AT ALL. The DP does all this. That definition you read is pretty spot on. The director directs the ACTION. He is responsible for the overall product (outside of the producer, but imagine the director as the artistic piece, and the producer is more money and logistics). So the DP works under/with him to determine the look of the film, and the DP actually captures it. On some REALLY large budget films, the DP may not even always handle the camera, he may just be the visual director - you may have someone else using a Steadicam, G&E might rig it to something, you have other camera operators, etc. But the director will work with him on how to frame, light, capture the scene, and then the director guides the actors through the scene. The director directs all motion, basically. So, if I'm acting in a scene, first thing the lighting is set up, which G&E (grip and electric, sorry, you probably don't know that one) does in accordance with what the DP and director want. Once the scene is lit (this is way after a PD has decorated the set), props may be moved into position the actors will be using or interacting with. Actors are then put on marks and the director may walk them through their movements for the scene. Blocking is done (if there is motion, where is the actor ending up and facing - this relates to how the camera will end up capturing their motion). The director walks the talent through all this. He may or may not even be the one calling action then. It's the first assistant director, that after blocking is done, will ask for camera to be ready and roll, then sound, and then will call for action. The director typically calls cut. The director will often work with the editor and sound mixer, colorist, etc. in post production to determine the final look and feel of the film and make sure it jives with his vision for the project. On independent projects, the director will typically have "final cut," which means he decides when the film is ready to be distributed in terms of content. At a studio level, a suit may have to say when it's okay to be released. And while I've chosen to use male pronouns for grammatical simplicity and consistency, knowing the plight of women directors in the industry, I simply want to say no disrespect.