Relationship between Turing Machine and Lambda calculus?
The lambda calculus is older than Turing's machine model, apparently dating from the period 1928-1929 (Seldin 2006), and was invented to encapsulate the notion of a schematic function that Church needed for a foundational logic he devised.  It was not invented to capture the general notion of computable function, and indeed a weaker typed version would have served his purposes better.

It seems to be incidental to the purpose of that the calculus Church invented turned out to be Turing complete, although later Church used the lambda calculus as his foundation for what he called the effectively computable functions (1936), which Turing appealed to in his paper.

Church's simple theory of types (1940) provides a more moderate, typed theory of functions that suffices to express the syntax of higher-order logic but does not express all recursive functions.  This theory can be seen as being more in tune with Church's original motivation.

References


Church (1936).  An unsolvable problem in elementary number theory.  American Journal of Mathematics 58:345—363. 
Church (1940).  A formulation of the simple theory of types.  Journal of Symbolic Logic 5(2):56—68. 
Seldin (2006). The logic of Curry and Church.  In Handbook of the History of Logic, vol.5: Logic from Russell to Church, p. 819&mdash;874.  North-Holland: Amsterdam.


Note This answer is substantially revised due to objections by Kaveh and Sasho.  I recommend the Wikipedia timeline that Kaveh suggested, History of the Church–Turing thesis, which has some choice quotes from seminal articles.