Okay, here's how I understand it.
On the skills tab of your character sheet, there are two areas to look at here. These are: Weapon Types, and Weapon Specializations.
Weapon Types (from memory):
Two Handed Piercing
Two Handed Blunt
Two Handed Slashing
One Handed Blunt
One Handed Slashing
Hand to Hand
Archery
Weapon Specializations (also from memory):
Spear and Javelin
Bow
Crossbow
Staff
Mace
Hammer
Club
Greataxe
Greatsword
Greatmaul
Longsword
Shortsword
Hand to Hand
Weapon Types:
skill cap = (your level) x 10.
Weapon Types represent your general skill in a category of weapon, like slashing or blunt. They are also known to be the skills that increase the slowest. You can max all of these skills, meaning you have enough points to do so, however, it will take a lot of work.
Having these skills maxed for your level gives you a 95% chance to hit on a swing with a weapon of that type. So obviously, the ideal situation is to have them all maxed, giving you a 95% chance of hitting for every weapon type.
The best way to level these skills up is simple, through use. However, to help speed the process along, I recommend duel wielding two of the desired weapon type if it is one-handed, and using them against lots of even con or higher targets. Increasing weapon skills works just like gaining adventuring experience. You get more experience from a target that is higher level than you, and similarly, you have a higher chance of increasing your skills against a target that is higher level than you.
You should first figure out what kind of weapons you want to use the most. As an example, I'm a Warrior, and I find myself using primarily my longsword and shield in tanking situations. Thus One Handed Slashing is my first priority every time I level. Once that is maxed, I switch to One Handed Blunt. After that, I move on to the Two Handed weapons, starting with slashing, etc. Archery also rises since I use my bow to pull a lot.
So, raising these skills is just a matter of use, and the order in which you raise them is a matter of your personal taste, as well as what weapons work well with your class.
Weapon Specializations:
skill cap = (your level) x .5
These skills are more simply explained than Weapon Type skills. They also go up through use, but they are split into the specific weapon types. Thus, under the One Handed Blunt weapon type, you have Hammer, Mace, and Club. Unlike Weapon Type skills, however, you can NOT max all of these skills. With each level, I believe you get 30 points to allocate among the different weapons. This means at most you can keep 6 weapon specializations maxed at once. Again, assign them as you see fit, according to your fighting style and class.
These skills increase more quickly, so in order to make sure your points go to the correct skills, making use of the lock buttons on the right side of the window is recommended for the skills you do not want points to go into. Again using myself as an example, the weapons I currently tend to use are a longsword, a shortsword, a hammer, and a Two-handed sword. So, I max out those four skills first. In addition, I'm using a bow, so Bow maxes as well. That leaves me 5 points to spend somwhere. I usually just spread them around, spending the 5 on a different skill each level, so that they don't fall way behind. Thus I have those 5 maxed, while my other skills, are high enough for me to use the weapon effectively, but not maxed.
The benefit of maxed weapon specializations is a Damage Bonus when you are using that type of weapon.
You can obviously use a weapon if your skills are not maxed, but you will be less effective than you can be. However, it is important to note this, because even if your greatmaul skill is not maxed, you can still use it to skillup your Two-Handed Blutn weapon type skill.
Hopefully this answers a lot of questions people have. If there is still any confusion, or something folks would like me to add, let me know and I'll do so.
I also should have put this in the skills forum, but it was pretty spur of the moment, sorry.