Seemingly simple, yet interesting and complex question. There is no physical test that will diagnose it, as there is for other diseases. And the idea that it's actually a disease is debated constantly, with no clear winner. Many people agree that the only person who can accurately diagnose it, is the person experiencing it. Yet many people have lots of reason to know beyond the shadow of a doubt that someone they know or love is indeed an alcoholic. In AA, people define it through the big book, and there's a paragraph or 2 that describes what the book calls a "real" alcoholic. But many take issue with that, especially those who don't fit the profile, yet know they couldn't stop drinking.
There are also lots of questions as to whether there are different levels of alcoholism. Can someone be more alcoholic than someone else, or is it like pregnancy? Either you are, or you aren't?
I obviously have no answers to this stuff. At least none that couldn't be argued.
For me, I know that alcohol affects me differently than my non alcoholic family and friends. And once I start drinking, the compulsion to keep doing it is insanely powerful. Powerful enough to land me over and over again in horrible trouble, physically, mentally and spiritually. That's all I really need to define it for myself.