To say your body makes or produces fat is somewhat of a misnomer. At birth the number of fat cells (called adipocytes that specialize in fat storage) in your body is a relatively fixed number. The same can be said about many other types of cells in your body, including how many muscle fibers you are born with. You don't "grow" new muscle fibers when you adopt a weightlifting regimen; the fibers you do have, however become bulkier resulting in a larger, more muscular appearance. The equivalent is true about adipocytes; your body doesn't make more, it just fills the cells you do have as you store extra fat.
A few studies have indicated that some morbidly obese individual's bodies have shown the ability to generate new fat cells once they have filled all the ones they were born with, in a rare process called lipocytogenisis. There is a maximum volume that can be filled by fat cells even though they are elastic structures that easily shrink and expand, based on energy requirements and needs.
Deposition of fat into adipocytes is a relatively uncomplicated process conceptually. When you eat a sandwich for example, your body breaks each of the food particles into smaller components as, carbohydrates, protein, and or fats. The energy is used to fuel the basic metabolic processes of the body like breathing and heartbeat, to more complex processes of large muscle movement during exercise. The body derives the majority of its energy sources from the breakdown of carbohydrate and proteins and stores the energy in the form of glycogen. Glycogen is a sugar molecule that is primarily maintained in muscles to be used as a short term energy source that is easily accessible by the body.
While fats can be utilized as a primary energy source in very active people and athletes, it must first go through a more extensive process and be broken down by a process called beta oxidation, which prepares the fat molecules to be used in other pathways to be utilized as energy. Everything your body takes in and cannot utilize or store is excreted as waste product.
The lipid energy that is stored in adipocytes and forms adipose tissue is histologically classified as loose connective tissue. A healthy amount of body fat that helps form adipose tissue is required to maintain normal bodily functions, and also serves as a means of insulation and protection to organs. Only when there is a caloric imbalance and your body is burning fewer than it takes in, do your lipocytes begin to capture and store extra fat in adipocytes and a person begins to accumulate adiposity in the form of excessive body fat.
The consequences of excessive adiposity are far reaching and result in the individual becoming obese and suffering from associated co-morbidities referred to as Metabolic Syndrome that include: insulin resistance, high blood pressure, dyslipidemia, and cardiovascular disease. Understanding how fat is stored as a consequence of calorie imbalance can be important in keeping your levels of body fat in acceptable and healthy ranges.