When someone says they are 'tipsy,' this generally means to say that they have had a drink or two and they have developed a nice, pleasant buzz. The person is just starting to feel the effects of alcohol, and they are having fun. The person still can walk and talk, and their judgment is still intact. When someone becomes drunk, they will generally slur their words, they may see double, or they may even sway or stumble from side to side. Sound judgment and driving skills are starting to become impaired at this point.
The person may also have an encounter with the opposite sex, which they may either regret the next day or not remember very well. Saying that someone is a drunk means that they are a habitual drinker, and it is having a negative impact on their life. They may be at the point that their drinking is affecting others, and they may need help.