The name Smalltalk usually refers to ANSI Smalltalk or Smalltalk 80 (of which the first is based on). While most implementations are close to the standard, they vary in different aspects (usually referred to as dialects).
Each implementation has it's own method of installation.
Squeak (Windows/Linux/Mac OSX)
GNU Smalltalk (Windows/Linux/Mac OSX)
Dolphin Smalltalk Originally commercial, now free open source. (Windows only)
Cuis Smalltalk A Squeak fork with a focus on reducing system complexity.
VisualWorks/Cincom Smalltalk Free trial available.
VisualAge Smalltalk Originally by IBM, now Instatiations. Free trial available
Smalltalk/x (Free for personal use?)
GemStone/s Free community edition available.
Amber Smalltalk A Smalltalk that lives in the browser.
Redline Smalltalk Smalltalk for the JVM.