Knowing the style will enable me to go and find tutorials to help me get started.
I would say it falls in the "lightwave" and/or "abstract" category. As mentioned below, "smoke" as well.
I would just type "Smoke Effect Illustrator Tutorial" into Google. You can see a similar video tutorial Adobe Illustrator Brush Tutorial - Vector Smoke
The basic idea is to create the outer edges of the smoke and then use create -> blend to make the inside lines of the smoke.