Weapon appraisals are broken down in the following way:

    >draw sabre
    You draw out your silver-edged sabre from the staff harness.
    >app sabre
    It is a medium edged type weapon.

    You are certain that it could do:
      heavy puncture damage
      low slice damage
      low impact damage

    You are certain that the sabre is fairly balanced and is fairly suited to
    gaining extra attack power from your strength.

    You are certain that the silver-edged sabre is moderately strong, and is in
    pristine condition.

    The silver-edged sabre is made with metal.
    You are certain that the silver-edged sabre is worth exactly 520 lirums.
    Roundtime: 8 seconds.
  

This format tends to be written in short hand when speaking of appraisals as h/l/l f/f, meaning heavy puncture, low slice, low impact, fairly balanced and fairly suited to strength.

What does it mean?

Every weapon (and armor) has a certain set of statistics related to how much damage it can do, how well made it is, and how it interacts with your stats. With a sufficient amount of appraisal skill, one can find out what the general feel for a weapon is, but these terms are a bit vague and there can be quite a bit of difference between two weapons which appraise the same.

Weapon Type

First we'll simply note the item which you are told about the weapon to begin with, which is its type. Basically this just informs you what weapon skill you'll be learning (HE, ME, LE, Light Crossbow, etc) from using the weapon. In the case above, the sabre is a medium edged sword, so it'll teach the medium edged skill. At the end of this document, I will list a few common abbreviations for weapons and some examples of what weapons fall under which categories.


Damage

Secondly there's the damage section, which is listed in order of puncture damage, slice damage and then impact. These stats are important in finding out just what sort of style you should use when wielding the weapon. Above I have appraised a sabre, which is less common than the ubiquitous scimitar, which most people tend to think of when using medium edged weapons. In this case, the sabre is quite obviously biased towards puncture attacks, meaning critters which don't do well against puncture are going to get hurt quite a bit more than if they were attacked by a slicing weapon. A forged scimitar might appraise something like l/h/p r/f meaning that it has low puncture damage, but heavy slice, so it's more suited to slice attacks, rather than thrusting. Blunts tend to be biased towards impact, so they'd have appraisals more in like with l/l/h d/w meaning bash, draw, sweep and such are more effective than jabbing.
The damage codes from lowest to highest are as follows:

none, poor, low, fair, moderate, heavy, great, severe, bone-crushing, devastating, god-like

So, a weapon with poor puncture damage just isn't going to do as much damage as one with moderate or heavy. It's impossible to really put numbers on what kinds of damage you can expect with moderate vs. heavy, but the differences are substantial in combat. I should also take this time to note that unless you get a certain on your appraisal, there's no guarantee that it's correct.

Suitedness and balance

Thirdly, there's the suitedness and balance of a weapon. These are denoted above by the fairly balanced and fairly suited section. This section, until recently was always a bit iffy, however, with new fixes implemented in the world, balance of a weapon has become fairly important. Basically, the balance plays off of your agility when determining how easily you're going to hit a critter. This shouldn't be confused with how many weapon ranks you have, as that will always be more important, but a weapon that has reasonable balance will hit more often, or perhaps just better, with the same ranks in the weapon than one with poor balance. Agility, balance of weapon and ranks all play a major role in how well you hit a critter. The balance is more important to making contact than actual damage, however, as that is covered more by the suitedness to strength. This suitedness is covered by the second part of that app, saying that it is fairly suited to gaining extra attack power from your strength. Much as balance plays off of your agility, suitedness plays off of your strength. The stronger you are, the harder you hit, but your weapon also needs to be made to take full advantage of your strength, needless to say, ranks in a weapon have a major impact on damage as well.
The stats on balance and suitedness follow along the same lines as damage, but use different adjectives. I may be wrong on some of these:

not, dismally, poorly, fairly, reasonably, well, excellently, superbly, incredibly, unbelievably

Again, what we find is that the better suited and balanced a weapon is, the better it is. Most store bought weapons are in the poorly and fairly area and are just fine. Reasonably is generally the best you're going to get for swords short of claymores, greatswords and the mod slice or impact katars made by a few forgers.

Construction

Now there's the construction of the blade and its current state. The construction of a weapon is of some importance as the weaker the construction, the more likely it's going to break during combat. Any weapon below moderately strong has a pretty high chance of getting scuffed up after a while and a very high chance of being broken while hunting critters such as guardians and granite gargoyles, as well as critters which parry. The latter group of parrying critters tend not to be as dangerous to your weapon as gargoyles are, but can still pose a problem. Moderately strong and well constructed weapons are almost immune to such breakage but it never hurts to appraise them after you've hunted gargs just to make sure. Again, this section I'm not too sure of the scale, but it's something like this:

extremely weak, somewhat flimsy, average strength, fairly sturdy, moderately strong, well contructed, very strong

I don't have a list of what the states of disrepair are, but the rule of thumb is that if your weapon isn't appraising as in pristine condition, go get it repaired as it's less effective and more prone to breaking beyond repair.

Made with

Next is what the weapon is made out of it, which is fairly unimportant, except to tell you who can repair it. Metal shops do weapons made of metal, leather repair shops tend to do things which aren't metal. Also would influence dying at the Leth dyer or in Knife Clan.

Price

The cost of the weapon is, in my opinion practically of no value. The only thing it truly means is how much it is going to cost to get repaired. The more expensive the weapon is, the more it costs to repair. There is very little direct relation between its cost and how good it actually is.