DMAIC is an acronym for problem solving in the Six Sigma process. It stands for Define-Measure-Analyze-Improve-Control. The pronunciation is Duh-May-Ick.
While Six Sigma = DMAIC for many people, there is another improvement method that also uses DMAIC: Lean. If you look at the kaizen process, you will notice that it follows nearly the same steps as the DMAIC methodology. Other problem solving methods have similar approaches as well. For example, the 8D process very closely parallels the DMAIC methodology.
Why DMAIC? It provides a systematic way to approach a problem. Using the DMAIC methodology to structure an effort to solve a problem helps prevent skipping steps, which can lead to poor solutions.
While is seems foreign at first, when you look more closely at it, the DMAIC methodology follows a logical flow. In fact, there is a good chance you already follow these steps when solving problems on your own. You probably come up with some sort of problem definition, research the problem, dive into the details, and then come up with a solution. DMAIC simply formalizes what you already know.