In the section on acids and bases, you learned that acids are substances that elevate the concentration of protons in a solution, and a base reduces it. More specifically, Swiss chemist Svante Arrhenius formulated this definition:

(a neutralization reaction) or one that is there just because of the auto-ionization of water:

OH^-   +   H⁺   ⟶   H₂O

A weak base like ammonia raises the hydroxyl ion concentration by accepting a proton from auto-ionized water:

H₂O   ⇌   H⁺   +   OH^-

NH₃   +   H⁺   ⇌   NH₄⁺

So the Brønsted-Lowry definition of acids and bases is simple and encompasses most of the acids and bases we work with regularly. However, there are other compounds capable of changing pH that aren't so obvious, and another definition is required.

The F^- effectively donates a pair of electrons (outlined green dots) to BF₃.

Lewis' definition of a acids and bases is that they accept or donate a pair of electrons, respectively. In the BF₃ case, BF₃ accepts a pair of electrons, and therefore acts as an acid, and F^- acts as a base by donating a pair of electrons.

Lewis' definition is completely compatible with the neutralization reaction

H⁺   +   OH^-   ⟶   H₂O

if we look at where the electrons go:

Here the lone (acidic) proton accepts a pair of electrons from OH^-, and therefore acts as an acid in the Lewis theory, and OH^- is the electron pair donor – a Lewis base.

Lewis' theory is the most all-encompassing theory of what makes up an acid or base. It's not always easy to recognize components of a solution that will raise or lower pH if they aren't obvious H⁺ or OH^- donating compounds, but the Lewis definition helps. The Lewis definition is also handy for acid and base solutions that aren't aqueous, and many such solutions exist.

because ammonia is a Lewis base, so are all of the other group XV elements when they form compounds with three bonds, such as PH₃ (phosphine).