What is the difference between Arrhenius, bronsted-lowry and Lewis acids and bases?

In Arrhenius, we are limited to cases in which water is the solvent (this is assumed by Arrhenius). An acid will dissolve in water to produce #H^+# ions, while a base will dissolve in water to produce #OH^-# ions.

Bronsted and Lowry state that an acid is any substance that will donate a proton (meaning an #H^+# ion). This includes the case where the donation of the proton is made to a water molecule (which therefore includes everything Arrhenius would have considered), but allows for donation of protons to many other substances, opening the door for general acid-base reactions (proton-transfer reactions) such as

#HCl + NH_3 rarr NH_4^+ + Cl^-#

(Bronsted and Lowry also change our thinking of what makes a substance a base. All that is required is that a particle (atom, molecule or ion) be able to acquire a proton, and that particle is a base. Check out the role of #NH_3# above)

The Lewis definition goes this one further in stating that an acid is a substance that can receive an electron pair (meaning the lone pair of a particle) and that a base is a substance that can donate a lone pair.

Again, this definition includes all cases that fit into the B-L scheme, because the #H^+# proton Bronsted and Lowry refer to is a proton with an empty orbital. This orbital can bond with the lone pair of a particle such as the ammonia molecule (the N atom has a full orbital not used to bond to the three H atoms, hence a lone pair).

However, Lewis also includes cases in which the #H^+# proton is not the particle being transferred, and so broadens the concept of acids and bases to include many more cases. For example

#BF_3 + F^-# #rarr BF_4^-#

would not be an acid-base reaction according to Bronsted-Lowry, but does qualify in the Lewis sense.