Recently, I bought some Australian darts, and some of the points were loose enough that I was able to get a good look at them and figure out how to make them. They're easy to carve, fast to make, look primitive, and seem to be completely functional as well. What more do you need? The only tool I've used on these is a utility knife.
To start with, you just need a small blank of wood. The ones I've been making are pieces of pine, about 3 1/2" x 3/4" x 1/4".
Begin by shaving both sides to the edge, to make it more like a blade.
Decide which end will be attached to the dart shaft, and then cut down towards the end, leaving it blunt. You don't want this part sharp, or it will cut through the binding.
Turn the wood blank sideways and bevel the shaft end. Try for as shallow an angle as you can, because this is where it will be glued to the shaft, and the more contact there is, the better the grip will be.
Now take your knife and curve the tip until it comes to a point. Make it smooth and continuous, and thin the sides out as well so you have a good sharp point..
You're done! All you need to do now is prepare the end of the shaft by cutting a bevel at the same angle you cut the base of your wooden point, and then glue it down and wrap it with sinew. Hopefully your final piece looks something like this.
You can obviously make them longer, wider, whatever to fit your shaft better, and if you use a denser wood you'll have more weight at the end as well, which is always good. Good hunting!