Determination of whether a stamp is used or unused relies on it's cancellation. If the stamp has been cancelled it is used, if it has not been cancelled it is unused. Gum is irrelevant, some used stamps still have their gum and some unused stamps have no gum.
There is one exception to this rule and that is precancelled stamps. Precancelled stamps are stamps that in fact are cancelled before they are even purchased from the Post Office. Older pre-cancelled stamps usually will have the name of a city and state printed in black over top of the stamp design. Newer pre-cancelled stamps are generally imprinted with the type of rate they are to pay right in the design of the stamp. They may say "bulk rate" or "pre-sorted" or whichever rate the stamp is designed to pay. Stamp you receive on your "junk mail" are generally precancels. These stamps are not normally cancelled when they go through the mail stream, they remain uncancelled but are still used. In the case of precancelled stamps the gum is the determining factor, full gum means the stamp is unused, no gum means the stamp is used. Some pre-cancels do get cancelled in the mail stream and this of course makes them used.
Gum is an important factor in determining a stamp's value. Some collectors care very little about the gum side of the stamp, other collectors only collect stamps that have pristine gum. Even if you fall into the first category, you should be aware of the conditions that affect gum, there is no sense in paying NH prices for NG stamps.
Here are some common terms, including those just used.
NH - Never hinged - The stamp has never had a hinge applied to it. NH has an expanded meaning though, the gum must be in like new condition with no marks of any kind whether made by a hinge or by something else.
H - Hinged - The gum has had a hinge applied to it.
LH - lightly hinged - The gum has had a hinge applied, but the mark left is unusually small or light.
Hinge Remnant - The gum has had a hinge applied to it and a portion of the hinge was so difficult to remove that it was left in place, attached to the stamp.
Disturbed gum - The gum has been damaged in some way other than hinging. This can include fingerprints, glazing, bubbling or anything else that affects the gum.
No gum - The stamp is unused and has no gum.
RG - Re-gummed - The stamp has had new gum applied to it in place of the original gum. This is most often done to try and deceive buyers into believing they are buying an original gum, never hinged stamp. There is nothing wrong with the buying or selling of re-gummed stamps as long as both parties know the true condition of the stamp. Regummed stamps generally will sell at or slightly above the price of a NG stamp.
Gum skip - When the gum was applied during manufacture it was not spread completely over the stamp and it left a portion of the stamp without gum. It's usually a thin line of area left ungummed.
MNH - Mint or Mint Never Hinged - Mint means the stamp is in new condition, as it was when it was purchased from the Post Office. Many times you will see stamps listed as "Mint NH" or "MNH", this is a misuse of terminology as it is redundant. By saying mint, you are already saying NH as that is the condition it was in when it was purchased by the post office. Even though it's technically incorrect, MNH is still a commonnly used term.
Besides catalog value, this is the most important factor in judging a stamp's value to most people. Centering is how well the design of the stamp is centered within the perforations or edge of the stamp. I have included images of a stamp in each grade as I list the grades.
S - Superb The stamp design is very nearly perfectly centered within the perforations
XF - Extremely Fine - The stamp design is just towards the top, but still very nicely centered, almost perfect.
VF - Very fine - The design is further off from center, but still well away from the edge and very attractive.
F - Fine - The determination of fine is that the design can be very close to the edge, but it cannot be cut by the perforations
AVG - Average - The design is cut into by the perforations
There are quite a few different types of faults that can affect a stamp, all will reduce the value by varying degrees. How much the value is reduced is dependent upon a lot of factors - severity and type of fault, rarity of stamp and the disposition of the buyer and seller.
SP - Short perf - Short Perforation. This is when one or more of the perforation tips is not as long as it should be, but a portion of the tip is still present.
PP - Pulled perf - Pulled perforation. This is when a perforation tip is completely missing. A severely pulled perf can even mean some of the stamp has been taken away with the perf tip.
SE - Straight edge - This is when one one or more edges of the stamp do not have perforations. Some sheet stamps were made with straight edges on some sides of the sheets, other straight edges come from trimming the perforations off. Do not confuse a straight edged sheet stamp with a coil stamp that always has two edges without perforations, a booklet stamp that can have one, two or three edges without perforations, or an imperforate stamp that has no perforations.
RP - Re-perf - Re-perforated - This is an alteration made to the stamp to add perforations to one or more edges. Just like re-gumming it is often done by a dishonest person to try and improve the value of a stamp buy making it appear like what it is not. Re-perforations often occur on stamps with straight edges to make them look like fully perforated stamps. It is also often done to imperforate stamps to try and change them into a more expensive coil or sheet stamps with the same design.
Thin - A stamp with a thin has an area on the back where some of the paper has been removed. There will be a spot that is thinner than the remainder of the stamp. A thin can range from a very small speck, to as large as the entire stamp.
Gum bend - Gum crease - Gum wrinkle - This is a natural occurrence in many flat plate printed stamps. When the stamps were made the paper had a tendency to shrink. The gum did not shrink at the same rate so the stamps would wrinkle up. Most often on the issues where these wrinkles are common, they do not lower the value of the stamp. If they are severe, or if there are a lot on one stamp, they can then lower the value.
Face Scrape - A face scrape is where a portion of the front side of the stamp has been scraped away, leaving a spot in the stamp design.
Inclusion - An inclusion is a foreign piece of material that has been pressed into the paper when the paper was manufactured. It is normally a brown or black spot that can be on the front, back or in the middle of the stamp.
Tears - stains - creases - pinholes - holes - corner missing - missing pieces - These are other faults that can affect a stamp, they are just as they sound I do not think further definition is needed.