Who first introduced the pushdown automaton?
Ginsburg (in his book The Mathematical Theory of Context-Free Languages, McGraw-Hill, 1966) states in the Historical References (Section 2.7, page 81):


  Pushdown acceptors were first formalized by Chomsky [Ch5] and Evey
  [Ev], although the notion of a pushdown tape has been used since 1954.


[Ch5] N.Chomsky, context-free grammars and pushdown storage, MIT Res. Lab. Electron. Quart. Prog. Rept. 65, 1962.

[Ev] R.J.Evey, The theory and applications of pushdown store machines, Mathematical Linguistics and Automatic Translation, Harvard Univ. Computation Lab. Rept. NSF-IO, May, 1963.

The context can be read from the Preface of the same book.


  The concept of a context-free language was first introduced by Chomsky
  in 1959 in an attempt to find a reasonable mathematical model of
  natural language such as English, French, etc. In the period
  1959-1960, several papers developing the theory were written. In the
  late 1960, it was discovered that the "ALGOL-like" languages, that is,
  the languages defined by Backus normal  form (the metalanguage used to
  describe the widely publicized programming language ALGOL-60), where
  identical with the context-free languages.  Since then, there has been
  a flurry of activity in the theoretical development of context-free
  languages. Much of this work has been done by those concerned either
  with natural languages in connection with computers, or with
  programming languages. The remainder has been by mathematicians and
  logicians intrigued by the inherent problems, techniques, and results.
  This activity has been given birth to a number of theoretical results
  of concern to computer science, and especially to programming
  technology. For example, we have the characterization of a
  context-free language in terms of a pushdown acceptor (a device used
  in the parsing aspect of compiling). [...]