When working with materials in three js many of the materials support one or more types of maps for skinning the faces of a geometry, one such map is an alpha map. An alpha map is a gray scale texture where white areas of the texture will result in a face being fully opaque while black areas will result in the face being fully transparent. So then an aplha map will come into play when it comes to working things out with transparency in a three.js project along with the the transparency and opacity properties of a material.
When I first wrote this post I was using three.js version r104, and the last time I edited this post I was using r127. Three.js is still being developed and is moving pretty fast, in the future there might come another time where this code might break. So if things are not working out for you with this example, and many other examples on the open Internet the first thing you should check is the version of three.js that you are using.
So for a basic example of an alpha map in three js I have this example that makes used of a texture that is created from a canvas element. I just create a canvas and then make gray scale areas of the canvas by using the 2d drawing context. When drawing to the canvas any area that I draw as black will end up being totally transparent, and any area that is white will be fully opaque, shads of gray then set values between the two extreams.
I then used the THREE.CanvasTexture constructor to create a texture that I can then use with the alpha map property of a material that supports alpha maps such as the Mesh basic Material.
The transparent property of the material also needs to be set to true, and a renderer that supports transparency also needs to be used. The usual webGl renderer worked just fine fr me in this example.
There are a whole bunch of different maps to be aware of when it comes to skinning faces of a geometry with a material in three js. There is the plain old color map that can also be used, and it materials that respond to light sources there are many other maps of interest as well such as an emissive map. Still alpha maps are one ogf the many types of maps to be aware of, and they can be fun to play around with.