The tradition of printmaking in fine art is centuries old, though not all printmaking techniques are that old. A print is an original artwork created using whatever medium(s) and technique(s) the artist has selected. A print is not a reproduction of an existing artwork or painting.
A painting, drawing, or sketch may be used as the starting point for the print, but the end result is something different. For example, an etching made of a painting, something commonly done before the invention of photography and color printing processes. Take a look at these paintings and etchings by Lucian Freud and you'll quickly see how each is a unique piece of art. In traditional art printing, the printing plate is created by an artist by hand, inked and printed by hand (whether using a printing press or burnishing by hand, it's still a manual process, not computerized).
Why Bother with Printmaking, Why Not Just Paint? It's a bit like the difference between bread and toast. While they're very similar, created from the same materials, each has its own characteristics and appeal. Printmaking techniques may use paper and inks, but the results are unique and the process from start to finish quite different to painting.
What About Giclée Prints? Giclée prints are in a different category to fine art prints because they are reproductions of paintings, multiple versions of an existing painting for an artist to sell at a lower price. Though some of the conventions of printmaking are used by some artists for their giclée prints, such as the limiting the edition (how many prints are made) and signing the print at the bottom in pencil, they are reproductions created using an ink-jet printer from a scan or photo of a painting, not original artworks themselves.
Let's take a look at the conventions of How to Sign an Art Print...