Term: Cone
A Cone is pointed away from you in a quarter circle on a map or grid. It gets the name because, well, it's a cone.
When you aim a cone, the first square of that cone must share an edge with your space if you're aiming, as if you are moving at a diagonal angle. If you're Large or larger, the first square can run along the edge of the space you occupy (or at the nearest squares, if using a grid). You can't aim a cone so that it overlaps your space. The cone extends out for a number of feet, widening as it goes (if using a grid, every single 5 ft square vertically expands the cone 1 square horizontally, or 2 squares total).
If you make a cone originate elsewhere, use these same rules, with the first square of the cone using an edge or corner of that creature or object's space instead of your own.
Another way of putting it is that the cone is like a Line, but instead of a 5 ft width, its width expands from the user at a 45-degree angle. So a 15 ft cone will get 1 5 ft sq from the starting point, then 2 squares, then 3 at the end. As an example, if you cast the Burning Hands ability, you create a cone of flames that affects a quarter-circle area 15 feet on each edge. Every creature in that area, even if the edge is clipped, goes up in flames thanks to simple geometry.
If all of that is too confusing, here's a picture, done in Foundry.