Types of Acids and Bases:

Arrhenius: acids and bases dissolve in water releasing H+ and OH respectively. This is a narrow categorization.

Bronsted-Lowry: an acid is a H+ donor, and a base is a H+ acceptor. This is also a narrow definition, but a very important one.

Lux-Flood: an acid is an O2- acceptor, and a base is an O2- donor. This is an important definition in geological terms, where the chemistry of aluminosilicates is very important.

Lewis: an acid is an electron pair acceptor, and a base is an electron pair donor. This is the most general definition of the acid-base interaction, and covers all of coordination chemistry, and the HOMO to LUMO transitions.

Bronsted Acidity: The acid is a proton donor, and equilibria are set up in aqueous solution:

The acid is HF, and the base is H2O
The acid is H2O, and the base is NH3

Here we see that water can act as both the acid and the base. This kind of substance is called amphiprotic.

It should be noted that proton transfer is rapid in both directions, and the deprotonation of the HF or H2O, and acid-base chemistry in general, is best described as rapidly attained proton transfer equilibria.

Conjugate acids and bases:

In the first equilibrium above, H3O+ is a bronsted acid in the reverse reaction as it donates a proton to F. Hence, F is a bronsted base, as it is a proton acceptor. Similarly in the reverse reaction of the second equilibrium, NH4+ is a bronsted acid and OH is a bronsted base.

The general equilibrium can be written for the proton transfer reaction:

The species Base1 is called the conjugate base of Acid1 (eg. Base1 is F for acid1 is HF), and the species Acid2 is known as the conjugate acid of Base2 (eg. Acid2 is H3O+ for Base2 is H2O).

It should be noted that there is no fundamental difference between an acid and a conjugate acid or a base and a conjugate base: the conjugate base and conjugate acid are just the products of the reaction of the original acid and base.