A Brønsted-Lowry acid is a species that donates a proton, $\mathrm{H^+}$, and a Brønsted-Lowry base is a species that accepts a proton. An acid-base reaction is therefore a proton-transfer reaction.

Consider

$$\mathrm{HCl+H_2O\rightarrow H_3O^+ + Cl^-}.$$

Hydrochloric acid donates a proton to water, so $\mathrm{HCl}$ acts as the acid and $\mathrm{H_2O}$ as the base. The protonated water molecule $\mathrm{H_3O^+}$ is called hydronium.

When an acid loses a proton, the species that remains is its conjugate base. When a base gains a proton, the resulting species is its conjugate acid. Thus the reaction contains two conjugate pairs:

$$\mathrm{HCl/Cl^-}$$

and

$$\mathrm{H_3O^+/H_2O}.$$

Members of a conjugate pair differ by exactly one proton.

A reversible proton transfer can be written

$$\mathrm{HA+B\rightleftharpoons A^-+HB^+},$$

where $\mathrm{HA/A^-}$ and $\mathrm{HB^+/B}$ are conjugate acid-base pairs.

Water can play either role

In

$$\mathrm{NH_3+H_2O\rightleftharpoons NH_4^+ + OH^-},$$

water donates a proton, so here water acts as an acid while ammonia acts as a base.

A species able to donate or accept a proton depending on its reaction partner is amphiprotic. Water is a central example. The bicarbonate ion, $\mathrm{HCO_3^-}$, can likewise donate a proton to form $\mathrm{CO_3^{2-}}$ or accept one to form $\mathrm{H_2CO_3}$.

Acid and base are therefore roles in a proton-transfer reaction, not permanent labels attached to a molecule in isolation. The direction and extent of proton transfer depend on the relative acid-base strengths of the reacting conjugate pairs.