The term dry drunk refers to alcoholics who have stopped drinking alcohol. It means you believe the alcoholic in question will return to the bottle sooner or later, saying they're "a drunk whose bottle just happens to have run dry for now." It's generally used to deride those who have not sought help with quitting. Those who have sought help prefer to be called "Recovering alcoholics", or some other more respectful term. Calling someone a dry drunk completely denigrates any efforts they've made to change.
Aside from the derision, the term is actually descriptive. Most of those who try to quit drinking, start again eventually. Support groups or counseling improve the odds, because drinking tends to be a symptom of deeper issues which require counseling of one sort or another to work through. Even without counseling or support, on sheer force of will, some people can cure themselves and grow beyond their disease. But they're the exception to the rule.
The rest only deal with one symptom of their actual disease, so the desire to drink remains strong, it tortures them. Not drinking is a painful uphill battle they must begin anew each day. They do "not drinking", focusing on the avoidance of action, instead of doing some other activity that might help them cope. Speaking from personal experience, many become obnoxious and self-righteous about getting everyone around them not to drink in order to make it easier for them to avoid it. Focusing on other people's actions replaces their drinking in helping them avoid the source of their problems.
The behavior extends to all sorts of addicts, but seems most pronounced in alcoholics. Everyone around them can have "a couple drinks" and stop, while they cannot. Those that succeed only in "not drinking", remain visibly broken souls, uncomfortable in their own skin. These are the classic dry drunks. Unfortunately, most of us know at least one.