Lipids are extremely common in the body, and their use in biology tends to fall in several main categories: making up cell and organelle membranes, serving as a high density energy source, and operating as a class of signaling molecules.
Lipid molecules, because of their hydrophobic nature, spontaneously self-assemble when exposed to water in order to minimize their exposed surface area. Simple lipids like oils tend to form spheres called micelles; you have probably seen these blobs form when cooking or washing dishes. A more complex lipid type, called phospholipids, have both a hydrophobic region and a hydrophilic region. Such phospholipids can form bilayer membranes. Cells use these bilayer membranes as something like a skin, a barrier against the outside world. They also make up the membranes of organelles inside the cell, like the mitochondria, the nuclear membrane, the endoplasmic reticulum, and so on. Other classes of lipids, such as cholesterol, can be used to reinforce cellular membranes.
Lipids, particularly a class called triglycerides, are also a high density energy source in the body. Per unit of mass, fat contains over twice the net energy as sugars like glucose, so they make for an efficient storage medium. The downside of using fat for energy storage is that it metabolizes slower than sugar. The fastest, strongest muscles of your body, including your brain, run primarily on sugar because it metabolizes quickly, while slower muscles that require endurance run on fats.
Finally, certain types of lipids can be used as signaling molecules in the body. Particularly, this involves the class of steroid hormones. Because cell membranes are composed of phospholipids with a hydrophobic region, other hydrophobic molecules like steroids can pass right through them by diffusion. For instance, the hormone testosterone moves directly through cell membranes without having to be transported and so can have a direct effect on cellular functions within the cytoplasm of the cell.