As far as Squeak is concerned 'Smalltalk' has two meanings:
First meaning
The usual meaning of "Smalltalk" is either the Squeak programming language or the historical programming environment that Squeak derives from. Note that the "t" in Smalltalk isn't capitalized and there is no space between Small and talk.
Squeak is a descendent of Smalltalk-80. The core developers of Squeak (then named Squeak Central) in fact included many of the core developers of Smalltalk-80.
Squeak's language and much of the core class library is identical to that of Smalltalk-80: they both have objects, classes, single inheritance, blocks, garbage collections, collections, streams, model-view-controller, and many other bits. In the meantime the class library has been extended considerably.
ANSI Smalltalk is a recognized standard for Smalltalk. While Squeak is Smalltalk in spirit, it is not fully compliant with the ANSI Smalltalk standard. There is temporarily an add-on on SqueakMap which tries to move Squeak towards ANSI compatibility.