How does a Bronsted-Lowry acid/base reaction proceed?

Hint :Bronsted Lowry acid- base theory basically tells us that acid is the one who donates the proton and base is the one who accepts the proton and which gives conjugate base and conjugate acid as the product respectively.

Complete Step By Step Answer:
To explain it in simple way,
Acid $ + $ Base $ \rightleftharpoons $ Conjugate acid $ + $ Conjugate Base
Acid/ Base in aqueous solution, acid will donate proton and base will accept it.
To explain in detail, we will take one example: we will take $ HCl $ and add water into it.
 $ HCl\xrightarrow{{{H_2}O}}{H_3}{O^ + } + C{l^ - } $
 $ {H_3}{O^ + } $ is known as hydronium ion.
Here $ HCl $ donates its proton to water and becomes Bronsted lowry acid and another water molecule accepts the proton and acts as Bronsted Lowry base.

Note :
When acid donates proton the product of the reaction is conjugate base and when the base accepts proton it gives conjugate acid as the product from the reaction. According to this theory only one who accepts the proton is base. Don’t get confused between a Bronsted-Lowry acid/base theory and Arhenius theory.