Something extempore is spontaneous and unscripted. An extempore dance party on the city bus might be fun, while an extempore speech in front of the whole school could be terrifying.
The term exetempore comes from the Latin phrase ex tempore, "out of time," and was originally used to mean "without time to prepare." These days it usually comes up when a performer is improvising or a speaker goes off script or doesn't use a teleprompter: "The President's extempore remarks made his advisors nervous." You can also use the synonym extemporaneous.