MDMA is often referred to as a "love drug" as its primary effect is to endow the user with great empathy for others. During the peak of the experience, and indeed for up to 24 hours afterwards, the user feels at peace with themselves and the world. They find themselves instilled with confidence and poise, even if they are a person with self-esteem problems. In fact, MDMA was used in psychotherapy up until 1985, when it was made illegal by the DEA. (Today some therapists continue their MDMA sessions underground, due to the many benefits it offers.)
I've heard that it lets you stay up all night.
It does generally instill the user with energy. "All night" is a bit of an exaggeration: the peak of the drug's effect generally lasts about two hours, but it may continue to give you energy (thereby keeping you awake) for up to six. Thus if you were to take a normal dose at midnight, you could expect to stay up until dawn even if you're not normally a night person.
Some of the subjective effects that users enjoy include enhanced hearing (music sounds better), vision (things look crisper and more attractive), and sense of touch (touching, and being touched by others, feels very good).
I heard it makes you want to have sex!
This is not true. MDMA makes users feel loving, not sexual. If you roll with a husband, wife, boyfriend, or girlfriend, you may find yourselves more inclined to have sex after the drug has worn off. MDMA brings you closer to those you love, so this is only natural.
Some users insist on taking unreasonably high doses of the drug, which generally spoils the effect. See the section on dosage for information about the correct way to use MDMA.