Diane has offered a good explanation, but I'll add some informative detail. The word "lithograph" refers to printing from a specially prepared flat stone. The image was drawn on the stone in a grease pencil and then acid-etched. Alternating rollers of water and grease-based printing ink were applied to the stone before each impression (print) was pulled. True runs of lithographs were usually short editions--maybe 50, for example. The fractional number refers to the individual print number within the edition. For example, the twelfth print pulled from an edition of 50 would be numbered "12 / 50". These are always numbered and signed in the margin by the artist. (If you see the artist's signature in the painting image, but not in the margin, that's called "stone signed" but it really means nothing. It's just that the artist's signature on the canvas was captured in the photograph or scan of the image.)
Occasionally, the artist would alter his original printing stone by adding lines or textures or even making a new stone with a new copy of the drawing (because the old stones wore down fairly rapidly). First impressions pulled after any of those circumstances would be artist's proofs.
These are misnomers now, because the modern editions of these reproduced paintings are printed with offset lithography. Stones aren't involved. Printing plates and rubber transfer blankets and mechanical presses are involved. The press operator might adjust the flow of the various colors of ink at first, but just until everything looks right to the artist. The individual impressions happen FAST. 5,000 prints could be made in an hour. The set-up copies are the true proofs. A certain overage will always be printed, just because the press operates really fast and because the number of set-up copies can vary. All of the overage can be signed by the artist and sold as proofs. Nowadays, there is really nothing whatsoever that makes these copies more valuable than the numbered edition. Sometimes the proofs are sold directly by the artist, whereas the numbered prints would be sold by a publisher. In that scenario, the proofs might be a little more valued by some, but there is NOTHING inherently more valuable about them.
That all applies to lithography.
Giclee is a spray technique, as has been explained. Think of it as a high class ink jet print. The technology is highly valued by artists because we don't have to print a full edition at a time, as we did with litho. For example, we could print one or two or five or fifty prints at our own discretion and as they sell. We don't have to invest in a few thousand up front. I do giclees of my own work. I publish very limited editions of up to about 25 copies. I might actually print only a dozen or so copies at a time. Sometimes fewer. Large format giclee printers with multiple ink heads can be expensive to operate and they're slow, too. Thus, the higher cost to the buyer.
Then there's digital printing, which artists don't typically use, but which is used all the time for brochures and posters and other pieces of COMMERCIAL art. You can think of digital printing as high class laser copying or printing--usually on a fairly large scale. The beauty of this technology is that you can print...say...50 full color posters and then be done. With an offset litho press, you'd have to commit to at least 250 - 500 copies of anything to make it a practical job and sometimes you just don't need 500 copies of a thing.