What's particularly good about this plant is that it lends itself to many different styles and gives you lots of freedom to explore your creativity.
If you want a detailed intro into bonsai-ing your azalea, you can have a look here.
However, I want to cover the most common styles used for this flower (and ones I have tried myself):
This technique is not too difficult to execute, but will take a fair amount of time. Plants that don't have a natural tendency towards long roots will not be suitable specimens for this method. The idea is to have the azalea's roots grasping onto a rock such that the latter offers stability to the tree.
This one is similar to the traditional cascade style, with the exception that the end of the overflowing branch stays above the level of the pot bottom. This style brings out string roots in the azalea as the plant will need to cling harder to the ground to remain stable.
The idea is to have the branches of the bonsai growing on one side of the trunk and also have the trunk itself lean in that same direction. This gives an impression of the azalea having grown in a windy place with the wind constantly pushing the tree to one side. You can get an idea of how it looks on Ramesh Sinha's blog. Imagine having all those beautiful blossoms stick out on one side of the plant.
The tree is more or less upright, but doesn't have a straight trunk. The trunk is kind of curved in a zig-zag pattern and the branches stick out on every turn of the trunk.
The azalea bonsai can be styled into these and even some other styles and in this sense is a very flexible species.