Full Idea
Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack.
Gist of Idea
Cantor's sets were just collections, but Dedekind's were containers
Source
report of George Cantor (works [1880]) by Oliver,A/Smiley,T - What are Sets and What are they For? Intro
Book Reference
'Metaphysics (Philosophical Perspectives 20)', ed/tr. Hawthorne,John [Blackwell 2006], p.123
A Reaction
As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets.