Ecstasy (MDMA) is not physically addictive -- meaning, a person who uses it on a regular basis will not suffer physical withdrawal symptoms when they quit. They won't get sick, the way an alcoholic or a heroin addict gets sick when they suddenly stop using.
However, like most drugs, it makes you feel really, really good, so it can be psychologically addictive: as with most drugs, regular users can build up a tolerance to it. Ecstasy works by releasing all of the serotonin (the chemical that controls your mood and sense of well-being) in your brain at once. This flood of serotonin is essentially what produces an Ecstasy high.
Since Ecstasy depletes your serotonin level, many people feel depressed, anxious, and generally unhappy for a day or so after using Ecstasy. This is called a "comedown." There is no way to replenish your serotonin level, besides waiting several weeks for your brain to fully replenish the serotonin by itself.
Users who cannot, or will not, wait that long to do Ecstasy again find that the drug doesn't have the same effect as it did before. Because their serotonin level is already low, the high is not as good as it was when their serotonin level was normal. So, they must take a double dose, or even a triple dose, to get the same high. If they continue on this way, they will eventually have to take massive doses of the drug, just to feel normal. This is what's known as a psychological addiction.