What are keen weapons and how do you know which weapons are keen for which races?

Weapons are keen for all races. Basically, they improve the chances of a critical hit. I'm not sure of the specifics, but I think that a keen weapon that might initially only be a crit threat on a roll of 20 might be improved to a 19 and 20.

Yes, a keen weapon has twice the chance of scoring a critical hit than that of a non-keen weapon.

for example a rapier has a critical threat range of 18-20 if that weapon is also keen the threat range becomes 15-20 (improved critical would change it to 12-20) and if you have a weapons master (Prestige class from HotU) at high enough level the threat range will drop even more.

Yes, I'm quite tempted to try a dual scimitar or rapier wielding Weapons Master in my next HoTU game. If a scimitar or rapier is the weapon of choice, and improved critical is taken and a keen weapon is wielded, the weapons master will hit for criticals on rolls of 10 and above, and the damage multiplier would be *3. This is great!

the only problem with dual wielding rapiers and scimitars is that they don't count as light weapons for the purposes of two-weapon fighting and ambidexterity ie you still have serious penalties since they're "non-light" weapons.

If you can get your hands on a nice set of Kukri, there are some out there that are "Keen", they are very nasty indeed when dual wielded, and you keep the small weopons status.

Using a non-light weapon in the off-hand only gives you a -2 penalty to both attacks, so it's not that bad. Until you meet enemies immune to critical hits.... :D