How can an object (meteroid, vehicle) bounce off the atmosphere?

An object, initially in an eliptical orbit, has a radial (due to gravity) acceleration. If there was no atmosphere it would continue following that orbit, unless if it intercept the earth surface.

When the object enters in the atmosphere, the acceleration changes due to drag and bouyance.

$mathbf a = mathbf g + mathbf a_d + mathbf a_b$

$mathbf a_b = -frac{rho_{air}}{rho_{obj}}mathbf g$.

$mathbf a_d = -k|mathbf u|$, where $mathbf u$ is the velocity.

The differential equation for the motion is:

$$frac{d^2mathbf r}{dt^2} = mathbf g - mathbf g frac{rho_{air}}{rho_{obj}} - k|mathbf u|$$

Because of the buoyance component, (what decreases the radial downwards acceleration), the speed of the object can be too big, and the object goes to an outer orbit. On the other hand, the drag reduces that speed.

The net effect is a balance of that 2 opposing effects.

A simple analogy based on the same physics. Have you ever bounced pebbles on water? If not see this video .

Here is an explanation for the pebble, from which the angle and speed is the basic for the skipping meteorites.

At high speeds and small angles the effective density of matter the meteorite scatters off is high enough to be like the water surface to the pebble.