[Shea]

In this expression what is acting as the Bronsted-Lowry acid and the Bronsted-Lowry base? What is acting as the Arrhenius acid and the Arrhenius base?

HCO3- + H2O <--> H3O+ + CO3^2-

I need some serious help on this one...

[english]

HCO₃^- + H₂O «----» H₃O⁺ + CO₃^2-

If I say this right, your Arrhenius acid should be bicarbonate (HCO₃^-).  There is no Arrhenius base.

Bicarbonate is your Brønsted-Lowry acid and water is your Brønsted-Lowry base.

Look at the products.  What is donating and what is receiving the proton?  How would you get from
HCO₃^- to CO₃^2-?

How would you get from H₂O to H₃O⁺?

Think about this and you will understand where H⁺ is going.  Wherever the proton has to come from will be your acid, wherever it ends up will be your base. (I state this in simple terms for your understanding; technically, the hydrogen on bicarbonate is being attacked by an electron pair on water)

You can also roughly estimate relative pK_a values to see which acid is stonger.  As a general rule, in most cases water will be the weaker acid and weaker base.  The stronger acid of the two reactants will act as your acid, the weaker of the two will act as your base (water is the weaker acid of the two, and thus acts as a base in this reaction).