These words have very specific meanings in medicine, but can have slightly different meanings in casual conversation outside of medicine.
In the US, obese and overweight are terms used to describe people who weigh more than they should, according to current medical knowledge. The exact weights can change as new medical research is done, but an obese person is a very overweight person. Medically, fat is the type of tissue that makes a person overweight or obese.
In casual conversation:
Usually obese means very overweight, but not always. Both terms are used to describe someone who weighs to much when the speaker isn't trying to be insulting. Casually speaking, fat can either mean the tissue causing you to be overweight ("I've got a ton of fat on my belly!"), or it can be a mildly insulting description of someone who weighs too much.