Giclee is a french word that doesn't translate exactly into english. As close as you can come is " spray, mist or splash"
Most Giclees are produced on canvas but some are still produced on archival, uncoated paper, creating more of a water color effect than an oil painting.
In the last 10+ years Giclees have been produced on high-end, multi-head, large format, inkjet machines, capable of repproducing images of 40" X 200' with some now going to 6' X 200'.
Artist proof used to refer to a "signed off for approved color" by the artist for the pressman to use as a target, or sample, to run to during a traditional printing press "litho run". This signed sample was usually kept afterwards by the artist for comparison purposess to assure himself that the press run stayed true to color for the entire press run. Sometimes these were later sold as "one offs". I only mention this because some here may own lithos and artist proofs from the past and I didn't wish to confuse. Present day the term usually means the artist has inspected the Giclee for accuracy to the original, enhanced with paint, and then signed the piece making it not an original but the next closest thing.
Limited edition means the Giclee has a specific run quantity, signed, numbered and most of the time has some authenticity paperwork to go with it.
You can have an Giclee or litho artist proof without it being a limited edition and vice-versa.