See Dedekind's Contributions to the Foundations of Mathematics.
Letting aside the issue of "priority" (that are not interesting), we can say in a nutshell that Dedekind's contribution was fundamental to support the early development of set theory :
In the context of his work on algebraic number theory [1871], Richard Dedekind introduced an essentially set-theoretic viewpoint, defining fields and ideals of algebraic numbers. These ideas were presented in a very mature form, making use of set operations and of structure-preserving mappings [Cantor employed Dedekind's terminology for the operations in his own work on set theory around 1880].
[In 1872] Dedekind published a paper in which he provided an axiomatic analysis of the structure of the set $\mathbb R$ of real numbers. He defined it as an ordered field that is also complete; completeness in that sense has the Archimedean axiom as a consequence. Cantor too provided a definition of $\mathbb R$ in 1872, employing Cauchy sequences of rational numbers, which was an elegant simplification of the definition offered by Karl Weierstrass in his lectures.
The Cantor and Dedekind definitions of the real numbers relied implicitly on set theory, and can be seen in retrospect to involve the assumption of a Power Set principle. Both took as given the set of rational numbers, and for the definition of $\mathbb R$ they relied on a certain totality of infinite sets of rational numbers (either sequences, or Dedekind cuts).
But while Cantor's works was (mainly) involved with transfinite arithmetic and the problem of the Cardinality of the continuum, Dedekind's work of 1888 dedicated to the theory of the natural numbers (Was sind und was sollen die Zahlen?) applies the set theoretic concepts to "elucidate" and develop a very fundamental piece of our "basic" mathematical knowledge : the natural numbers.
See :
- Richard Dedekind, Essays on the Theory of Numbers (Engl.transl.1901 - Dover reprint). Ref to Cantor (and Weierstrass): Preface to the 1st Ed, page 36 and Preface to the 2Nd Ed, page 41;
I. SYSTEMS OF ELEMENTS [page 44] :
In what follows I understand by thing every object of our thought. In order to be able easily to speak of things, we designate them by symbols, e.g.,
by letters, and we venture to speak briefly of the thing $a$ or of $a$ simply, when we mean the thing denoted by $a$ and not at all the letter $a$ itself. A thing is completely determined by all that can be affirmed or thought concerning it. A thing $a$ is the same as $b$ (identical with $b$), and $b$ the same as $a$, when all that can be thought concerning $a$ can also be thought concerning
$b$, and when all that is true of $b$ can also be thought of $a$. That $a$ and $b$ are only symbols or names for one and the same thing is indicated by the notation $a = b$, and also by $b = a$. If further $b = c$, that is, if $c$ as well as $a$ is a symbol for the thing denoted by $b$, then is also $a = c$. If the above coincidence of the thing denoted by $a$ with the thing denoted by $b$
does not exist, then are the things $a, b$ said to be different, $a$ is another thing than $b$, $b$ another thing than $a$; there is some property belonging to the one that does not belong to the other.
It very frequently happens that different things, $a, b, c, \ldots$ for some reason can be considered from a common point of view, can be associated in the
mind, and we say that they form a system $S$; we call the things $a, b, c, \ldots$ elements of the system $S$, they are contained in $S$; conversely, $S$ consists of these elements. Such a system $S$ (an aggregate, a manifold,
a totality) as an object of our thought is likewise a thing; it is completely determined when with tespect to every thing it is determined whether it is an element of $S$ or not. The system $S$ is hence the same as the system $T$, in symbols $S = T$, when every element of $S$ is also element of $T$, and every
element of $T$ is also element of $S$.
We can find quite verbatim the same introduction in every modern algebra or calculus textbook.
