A shooting star is just rock, ice or dust. A small, unremarkable bit of debris.
From the original, swirling plane of dust and gas that orbited the young, forming Sun, and the aeons of impacts between large bodies, there are now the planets and their moons. But there are also the legions of rocky or icy objects throughout the Solar System. Some lie between Mars and Jupiter, in the Asteroid Belt, some lie beyond Pluto in the furthest reaches of the Sun's gravitational field. These "meteoroids" range from the microscopic in size to over ten metres in width. Not all of these are left over from ancient cosmic smashes: many of the icy, frozen objects are left by comets as they hurtle towards the Sun on their vast orbits.
Sometimes a group of meteroids come close to Earth. This is because of the Earth's orbit taking it through groups of debris left by comets, or because one has become dislodged by the competing gravity fields in the inner solar system. It hurtles towards Earth. As it comes into contact with the atmosphere, the air in front of it is compressed, heating it up. This heat then affects the object, causing it to disintegrate and burn, or even boil. The compression is generated by the object's speed (at up to 40 miles per second) and pressure of the sudden, dense gas. Some of the debris left by comets is composed of ice, whereas other objects are solid rock.
It becomes a meteor: a streak of light in the sky. A shooting star.
Because most meteoroids are very small (pebble-sized), this process burns them up completely and nothing is left of them. The impact with the atmosphere destroys the object completely before it has come within 75 km of the surface of the Earth.
Larger objects can survive the pummelling they take in the atmosphere and reach the surface. They smash into the ground, usually making a crater. If there is anything left of the object after its impact, the remains are known as a meteorite. Often the impact will destroy the rock completely and vaporise it. There are nearly 40000 known meteorites around the world.
The shooting star phenomenon has other interesting aspects. A fireball is a very bright meteor, considerably brighter than the majority of meteors. Some meteoroids do not head down towards Earth but graze the atmosphere, causing the shooting star, and then emerge into space again.
There we have it: nothing to wish on, and nothing extraordinary about it. Shooting stars are just burning bits of rock. They are very beautiful, though, so it is tempting to make a wish when you see one...