It is said that helium is lighter than air. What does that exactly mean?
It means that the same volume of helium gas has a lower mass than the same volume of air.
If it is lighter than air,then there should be absolutely no trace of helium in earth's atmosphere(it should have flown into the space)
Well, the gravity of earth affects Helium, too, so there should still be some helium left. More importantly, helium is constantly produced by the alpha decay of radioactive substances.
But it is true that Helium is negligible in the atmosphere, only 5 ppm of the air are Helium.
Yet it is one of the most abundant element on earth?
No, it isn't.
How is this stored? If the balloons filled with helium rise in the air, why don't the helium storage tanks/vehicles rise?
The Helium we exploit is stored under the earth at the same places where also natural gas is stored. It doesn't evade for the same reason why the natural gas doesn't evade: Because it is trapped in the rocks. There's a considerable amount there because it had millions of years to accumulate (recall that it is constantly produced by radioactive decay).