H represents hydrogen atom (1 proton/1 electron)
H⁺¹ represents hydrogen ion (1 proton/No electron)
p⁺¹ represents free proton
We will make no distinction between H⁺¹ & p⁺¹
(H⁺¹ actually represents nuclei of 1p, 1p/1n, 1p/2n)

Since it is impossible to have free proton in water,

terms hydronium ion (H3O⁺¹), hydrogen ion (H⁺¹), and free proton (p⁺¹) are often used interchangeably.
 

Arrhenius Theory
Arrenhius proposed (1883) that ionic compounds dissociate and can become free ions acting as separate entities in solution.
ACID: Produces H⁺¹ in Water
BASE: Produces OH^-1 in Water

HCl_(aq)H⁺¹ + Cl^-1
HNO_3(aq)H⁺¹ + NO₃^-1
NaOH_(aq)Na⁺¹ + OH^-1
NH_3(aq)NH₄⁺¹ + OH^-1
(Although NH₃ does not contain OH^-1, hydroxide ions form when added to water)

The Arrhenius theory also allows acid and base to neutralize each other producing salt & water:
Acid  +  Base Salt + water
H⁺¹_(aq) + OH^-1_(aq)H₂O_(l)
HCl_(aq) + NaOH_(aq)NaCl_(aq) + H₂O_(l)

Arrenhius theory most limited of the 3 theories; it requires reactions be aqueous and applies only to substances producing H⁺¹,OH^-1,or H₂O.

Bronsted/Lowry Theory
Bronsted & Lowry proposed a new acid-base theory in 1923.
ACID:  PROTON DONOR
BASE: PROTON ACCEPTOR

When acid transfers proton it becomes conjugate base (CB); when base accepts proton it becomes  conjugate acid (CA). (See Conjugate Acid/Conjugate Base)

Since protons are always transferred in the Arrenhius theory, all Arrhenius acid/base reactions are also Bronsted-Lowry acid/base reactions.
Bronsted-Lowry theory expands Arrenhius theory to include any proton transfer (water not required).
 

Lewis Theory
In 1938 Gilbert Lewis proposed a more general acid base theory.

ACID: Accepts pair of electrons for sharing
BASE: Donates pair of electrons for sharing

The product of Lewis acid-base reaction referred to as adduct. The proton itself can act as Lewis acid. Lewis expands acid/base reactions to include many substances without H in formula.
F₃B+:NH₃F₃B:NH₃Explained by Lewis but not Arrenhius or BL
acid   base        adduct

:NH₃ Base because of lone pair
F₃B   Acid because of empty orbital