To extemporize is to improvise, especially without preparation. When you extemporize, you’re making it up as you go along.
In some situations, you’re prepared and know exactly what you’re going to do. In others, you have to extemporize. A stand-up comedian who makes up jokes on the spot is extemporizing. If a football play breaks down, the quarterback has to scramble and extemporize by coming up with a new play. A public speaker who throws away notes and takes questions is extemporizing. Jazz musicians often extemporize. When you extemporize, even you’re not sure what you might end up doing.