Each visible meteor represents a grain of extraterrestrial sand being vaporized as it plows into the earth's atmosphere. Meteor showers are generally associated with debris boiled off comets, which are loosely bound clumps of dirty ice. In that case when a comet comes close enough to the sun some of this ice evaporates and liberates a little of the dirt that was in it. The trail of debris is called a meteor stream. The Geminids are somewhat different than the usual shower, however.
Asteroids are much denser and more solidly constructed objects than comets.
They are made of rock rather than ice, so, it is a little odd that this one leaves a meteor stream behind. Perhaps 3200 Phaethon had a collision with another asteroid in the recent past and some debris was cast off in the process. Comets and asteroids both represent stuff left from the period more than 4 billion years ago when the sun formed and the planets congealed around it. All of this took about a hundred million years to happen, which is an eye blink compared to the time that has passed since. That difference reflects the different places where they formed.