Fat, or adipose tissue, is found in several places in your body. Generally, fat is found underneath the skin (subcutaneous fat), but can be found in other locations depending on the sex of the person. Females tend to store fat in the chest, hips, waist and buttocks creating a "pear" shape. Men tend to store body fat in their chest, abdomen and buttocks, producing an "apple" shape.
The main job of fat cells is to store fat. Fat cells have an incredible capacity to store fat with a 'full' cell containing almost 99% of its weight as fat! When people talk about weight gain, they are usually referring to an increase in body fat. Overeating results in increased fat storage as the body has more energy than it needs so it stores the excess nutrients (usually dietary fat) in fat cells where it can be called upon in times of food scarcity such as dieting. The simple explanation of why body fat is reduced when you go on a diet is that the body still needs to meet all its energy needs and if it is not getting it all from food, it will draw upon its largest reserves of energy - body fat.
As a person gains weight and the fat cells fill and expand, the cells can eventually multiply to allow more fat storage. This process should be fairly obvious as there appears no limit to how much fat humans can store.
Having lots of fat cells though doesn't mean a person is overweight; that only occurs when the fat cells become filled with fat and for that to occur a person needs to be eating more food than what they need to fuel their body.