* Burning Up: As they speed through the atmosphere, friction causes them to heat up and glow, creating the bright streak we see.
* Vaporizing: Most meteors completely burn up before they reach the ground.
* Small Bits: Some very small meteors might leave behind a tiny bit of dust.
So, the "shooting star" itself doesn't go anywhere after it streaks across the sky. It's gone, vaporized into the atmosphere.