According to the glorious revelation of IUPAC, whose Gold Book is the canonic reference standard for the definitions of terms in modern chemistry :
A molecular entity or chemical species capable of donating a hydron (proton) (see Brønsted acid) or capable of forming a covalent bond with an electron pair (see Lewis acid).
A chemical species or molecular entity having an available pair of electrons capable of forming a covalent bond with a hydron (proton) (see Brønsted base) or with the vacant orbital of some other species (see Lewis base).
Thus, IUPAC have adopted the satisfactory (though proton-infatuated) Brønsted-Lowry definition for the majority of situations, simply because it is convenient for daily use. In situations where this convenient model does not apply, we are directed to Gilbert N. Lewis' definition. Thus, acids and bases have been divided into Brønsted molecules and Lewis molecules. The two models are complimentary, and coexist easily in the complex minds of chemists, as they are useful for different situations.
Gilbert Newton Lewis
(1875 - 1946)
Gilbert N. Lewis, otherwise credited with the characterisation of such fundamentals as the covalent bond and the name "photon", is responsible for the theoretical framework which defines acids and bases without resorting to the discussion of proton traffic. Recognising the limitations of contemporary definitions, he wrote:
"When we discuss aqueous solutions of substances which do not contain hydroxyl [ion], it is simplest to define a base as a substance which adds hydrogen ion. . . . Since hydrogen is a constituent of most of our electrolytic solvents, the definition of an acid or base as a substance which gives up or takes up hydrogen ion would be more general than the one we used before, but it would not be universal."
Thus, acknowledging the value of proton-based definition in routine transactions of chemistry, Lewis was looking for a more general definition which would retain its validity in a variety of nonstandard situations, in the absence of solvents, and in the absence of protons.
In his 1924 work, he went on to offer this definition:
"We are inclined to think of substances as possessing acid or basic properties, without having a particular solvent in mind. It seems to me that with complete generality we may say that a basic substance is one which has a lone pair of electrons which may be used to complete the stable group of another atom, and that an acid is one which can employ a lone pair from another molecule in completing the stable group of one of its own atoms."
IUPAC, over 80 years later, have not made any substantial changes to any of this. The Gold Book holds forth with the following definitions of Lewis' concepts:
A molecular entity (and the corresponding chemical species ) that is an electron-pair acceptor and therefore able to react with a Lewis base to form a Lewis adduct, by sharing the electron pair furnished by the Lewis base.
A molecular entity (and the corresponding chemical species) able to provide a pair of electrons and
thus capable of coordination to a Lewis acid, thereby producing a Lewis adduct.
The "Lewis Adduct" these definitions keep referring to is actually just the combination of a Lewis base and a Lewis acid in a weird state of non-covalent attraction. The textbooks describe this "adduct" as some sort of transitional state which occupies a space somewhere between true ionic and true covalent bonding, but is more accurately neither.
To refer to such a strange bond, chemists usually resort to a single central dot, eg. Me3B·NH3.
Thus, in the Lewis definition, the reaction between an acid A and a base B would look like this:
A + :B → A-·B+
Here, the (:) semicolon represents the pair of electrons which are available for "donation" by the base.
The adduct is composed of an anionic A- which has received electrons, and a cationic B+ which has donated them. In this definition, the hydrogen ion itself is actually the acid- as a proton, it is "a molecular entity that is an electron-pair acceptor".
It is perhaps best not to abandon the use of the hydrogen ion. It is a useful concept, even though - as Peter Stewart put - it is a symbol for a metaphor. The convenience of this symbol, in spite of its gross electrochemical inaccuracy, has made it a convention of scientific literature. This specific issue is explored in some detail in a chapter dedicated to the use of H+ as a narrative device. In short, it's ok to continue to talk about hydrogen ions, provided one is aware that the concept is deeply flawed.
In the exploration of these concepts, one can tumble down many deep rabbit holes. If one wishes to pursue acid-base chemistry in greater detail, one may wish to begin with the enormous, detailed 6th edition of Advanced Organic Chemistry by Smith and March. Additionally, one may wish to pay homage to Prof. Martin Chaplin's site "Water Structure and Science", which is a beautiful temple dedicated to the marvel and wonder of H2O.