We chemists have three acid/base theories or definitions to help us conceptualize what we categorize as acid/base reactions. I will summarize each of them briefly.
Arrhenius theory is the oldest and most limiting definition:
Lowry/Bronsted is what we mostly use in intro and general chemistry classes. It is a broader definition. It is based on a specific donor/acceptor model:
Lewis theory is the broadest of definitions of all theories and is particularly useful/popular with organic chemists and various organic reaction types. It too is a donor/acceptor model:
We will tend to use the Lowry/Bronsted (L/B) definition as our definition. Realize that it covers everything that Arrhenius covers - meaning the same acids and bases in Arrhenius theory are also acids and bases in L/B theory. It's just that the L/B definition covers a few more cases - especially organic amines as weak bases because they accept protons from water. They themselves do not provide the hydroxide to make a base, but they do provide the proton acceptor to accept a proton from water, thus leaving behind a hydroxide which will raise the pH of the solution just like any base would.
So in the most generic way of showing a L/B acid-base reaction,
Notice how in this generic case, the two starting reactants are both neutral in charge (zero charge). After a H+ jumps ship off the acid and onto the base, the result are two ions - a cation (former neutral base) and an anion (former neutral acid). Without all the circles and colors, this would be shown like this:
And now lets write an actual real reaction with acetic acid (CH3COOH) playing the role of acid (HA) and good ol' water, H2O, playing the role of base (B).