Wooden swords, known as Bokken or Bokuto, are often used for training by practicioners of sword fighting. There are several reasons for this. One, a bokuto is much cheaper than a well made katana, which can cost thousands of dollars (even back in feudal Japan, a katana would be quite an expense). Also, it is much safer, a beginner is much less likely to chop off something important with a wooden sword. They are also useful for fighting duels where neither party really wants to die over it. Nonetheless, they can be very dangerous in the right hands.
Kendo, a sport of Japanese fencing, uses bamboo swords, which are somewhat different from bokuto. A bokuto is a single, solid piece of wood, usually oak or some other hardwood, wheras the shinai of kendo is several thin bamboo rods tied tightly together.
It is said that the great Japanese swordsman Miyamoto Musashi used only wooden swords in his duels after about the age of 30 (even though his oppenents continued to use real swords). He wrote in his famous The Book of Five Rings (paraphrasing a bit) that it takes 1000 days to learn the techniques of sword fighting, and 10000 days to master them. After this is done, one can easily win using a wooden sword, or even without holding a sword in your hand at all (he refers here to using your opponent's sword as your own, a theme that pops up throughout Five Rings).
Training with a bokuto contributes to strength and flexibility in the arms and wrist, cardiovascular health, and maybe even better mental health. And it's awfully fun.
Be warned that while wooden swords are pretty tough, they are not unbreakable. In particular, many of the ones you can buy through mail order are not even made of oak, but of various other hardwoods that won't stand up to abuse that well. In addition, even the oak ones will break: I recently snapped mine clean in half. Be sure to check yours for any hairline fractures or other signs of weakness before and after practicing with it.