Grounding or earthing is making sure you're stable and solid, rather than being unfocused with your head in the clouds (being "airy"). Centering is feeling totally present and within yourself.
It's hard to explain if you've been taught it long enough ago (like trying to describe how you walk). But it is a way of focusing your consciousness. Some people use a tree meditation, that is, imagining you're a tree and sinking your roots into the soil, in order to feel what "grounded" means. Feeling attached to the ground helps to ground excess energy, yours or that of people around you, rather that let it build up in you and make you flighty or tired or upset or all of the above. It's like a lightning rod.
I'm sure someone will explain it better! Good luck!
