A point cloud is just a bunch of points. Although it is sometimes useful to talk about point clouds in any dimensional space, we usually care only about 3D.
So you can impress your friends.
Suppose you wanted to make a cool-looking 3D computer model of your new convertible, to put on your web site. This will definitely impress your friends, unless you live in Palo Alto or Northwest Austin, where all of your friends own more expensive cars than you. You can do this by first making a 3D point cloud model of your car, and then turning this point cloud into a cool-looking 3D computer model of your car.
The best way is probably to take a laser range scanner which shoots laser rays at various points on the car and records the distance from the scanner to each point. This gives you a bunch of data points, each point representing a specific position on the car in 3D space.
I recommend entering a Ph.D. program in computer graphics at a university that owns a laser range scanner.
Actually, even top-level researchers don't know how to do this well. Researchers are trying to come up with computer programs that can take any point cloud and turn it into a picture with nice smooth surfaces. They also want these programs to run quickly. This problem of making smooth surfaces from a point cloud is known as surface reconstruction.
Those researchers who aren't interested in fancy cars, have many other reasons to care about point clouds. Here is a list of some areas where point clouds can be useful.
This page contains my summaries of some papers researchers have written about point clouds: