Many martial arts use some form of wooden weapon, so quite a few people on this forum have used or will use a weapon made out of wood.
Most of the time, you'll be fine by heading out to your local martial arts store and picking up whatever weapon you're looking for. However, there are times when you want to be particular about the quality of your weapon, especially if it is to be used in partner drills or any other sort of contact application.
The following are a number of types of wood and their qualities. This is by no means an ultimate list, and I will be adding to it as I learn more.
White Oak- One of three most common woods that I have encountered in the making of weapons. It's a pretty solid wood, though I wouldn't trust it for strong contact in particularly heavy weapons, such as Ieku. Excellent for shorter weapons, such as nunchaku or tonfa. When White Oak breaks, it can break into pretty jagged pieces.
Red Oak- Similar to White Oak, but a bit heavier. This makes it slightly more resistant to impact than White Oak. This is the second of the three most common woods. I have a Red Oak bo that has survived six months of heavy contact drills with no ill effects other than hundreds of little dents. This is also one of two woods that Shureido (the official Imperial supplier of kobudo weapons) uses.
Cedar- This is the third of the most common woods. Cedar is a wide-grained, light wood. STAY AWAY FROM CEDAR FOR ANY CONTACT APPLICATIONS! I cannot say this enough. Cedar is rigid but weak, so it snaps. Worse, when it breaks, it breaks into large, sharp pieces, so the worst case scenario of an incident can involve severe piercing injuries. Cedar is a pretty good wood for lighter practice weapons, particularly for children. This applies to other light woods, such as pine.
Waxwood- The exact opposite of Cedar, Waxwood is a beautiful, tight-grain, white wood. Despite how tight the grain is, Waxwood is actually pretty light. It is good for contact application, and it will not splinter. When it breaks, it usually breaks totally, and doesn't break into such spear-like points that Cedar and Oak can produce. One warning, Waxwood has a tendency to warp very quickly.
Ebony- Another very tight-grain wood. Ebony is anywhere from a coffee brown to a deep black. Ebony tends to be a heavier wood. African Ebony is even heavier than Asian Ebony. Like Waxwood, it's pretty resistant to impact.
Shijiya- This is the Japanese Beech tree. While I don't know much about Shijiya with regards to creation of weapons, I do know that this is the other wood that Shureido uses, and that it can stand up to contact training.
Ash- Similar to the Oaks, Ash is a moderately hard wood. It can handle quite a bit of impact. When broken, it can make some pretty sharp points.
Ironwoods- This group of woods includes Ipe, Bocote, Cocobola. They are extremely hard (about three times as hard as Ash or Oak). When they break, the break is usually a bunch of shorter points rather than one long point. They also tend to be pretty heavy.
Purpleheart- This is an absolutely beautiful wood. It is a deep color with a sort of purple sheen or glow. It can't really be described. Aside from its appearance, Purpleheart can also handle quite a bit of impact, making it a decent wood for weapons.
Bamboo- A large, fast-growing cane plant. Its primary strengths are its speed of growth (making for cheaper weapons), flexibility, and weight. If properly cared for, it can handle insane amounts of abuse for how light it is. Instead of breaking, it usually splits.
Composite Bamboo- This is how shinai are made. Strips of bamboo are tied together, reinforcing the weapon quite a bit. This can handle even more abuse. Shinai, when struck against a target, can bend at a 45 degree angle with no damage.
Pressed Bamboo- Basically Bamboo particle board. This is decent for weapons. It's primary advantage is the minimal impact on the environment. The amount of material in a Pressed Bamboo weapon can be regrown in a matter of days rather than years.
Rattan- If Bamboo can handle insane amounts of impact, Rattan is like Superman. Its weakness is that if the skin is carved away, the cellular structure inside can collapse on strong impact. Therefore, you can only work with whole lengths. Thankfully, Rattan can be shaped using consistent pressure, allowing the creation of mock weapons such as a Rattan naginata. Rattan is amazingly light and durable. Rattan and Bamboo weapons don't have unnaturally smooth surfaces. They have a more natural surface, with the little bumps that the plant itself grew with.
Fiberglass- Not a wood, but since so many martial arts stores sell them, I thought it important to touch on this. Fiberglass should not be used in any contact applications. It will shatter horribly, usually on the first strong strike. Similar to Cedar, it makes an excellent lightweight practice weapon. It is more flexible than Cedar, so it handles the incidental stress of swinging the weapon around a bit better. Further, because Fiberglass comes in a variety of colors, it is frequently used as an addition to a brilliant display in martial arts tournaments.