When you do something seriously, you do it earnestly or thoughtfully. If you seriously consider walking the Appalachian Trail, it means that you put real effort and thought into contemplating such a trip.
If your boss speaks seriously to you, you know she means what she's saying, and if you think seriously about what a fortune teller told you, it means that you really consider what she said your future might bring. You can also use seriously to mean significantly, or to a worrying degree, like being seriously ill with the flu. Informally, seriously also means very or truly, like when you are seriously hungry.