Horizontal Velocity
What do you think would happen if the skater had no vertical velocity, but just horizontal velocity? We have already seen that with only vertical velocity the skater goes straight up in the air. Guess what, one never gets off the ground.
To jump straight up you need to have some amount of vertical velocity, and to move only horizontally you need some amount of horizontal velocity. Thus, if you want to jump up and forward, or follow a projectile path, you are going to need both vertical and horizontal velocity.
Normally when a skater is jumping, or when a person is projecting some object into the air, we don't think in terms of the vertical and horizontal velocity. Instead we think about the angle and speed of projection. For example, you might report that take-off was at a 45 degree angle at 50 mph. This, however, is equivalent to specifying vertical and horizontal velocity, it is just a different way of reporting the information. In fact, there is a very easy way to get from take-off angle and take-off velocity (often called resultant velocity) to vertical and horizontal velocity. Formally this is called vector addition, but it is as easy as drawing right triangles.