If you look at the reaction below

hydrochloric acid + water hydroxonium ion + chloride ion
HCl(aq) + H2O(l) H3O+(aq) + Cl-(aq)

According to Arrhenius, hydrochloric acid is an acid
because it produces hydrogen ions in water
(hydrogen ions in water become hydroxonium ions).

According to Lowry and Brønsted
hydrochloric acid is an acid because it is a proton donor.
A proton is a hydrogen ion.
A proton donor is a substance which gives a hydrogen ion away.
If you look at the reaction above
hydrochloric acid gives a hydrogen ion to water.

A base is a proton acceptor.
This means that a base will gain a hydrogen ion.
Water is a base when it is put with hydrochloric acid
because water will gain a hydrogen ion to become H3O+.

acid + base acid + base
HCl(aq) + H2O(l) H3O+(aq) + Cl-(aq)

On the right side of the arrow, H3O+ is an acid
because it can give away a hydrogen ion to become H2O.
Cl- is a base because it can gain a hydrogen ion to become HCl.