Q:

What is the Sitz im Leben approach in Form Criticism?

What is the Sitz im Leben approach to Biblical hermeneutics? How is it used in form-critical exegesis?

A:

For OP: "Sitz im Leben" (German for "setting in life") is not itself an "approach", but is a fundamental component of "Form Criticism".
Sitz im Leben and Genre
I would like to add one caveat lector to the otherwise helpful answer from Richard. Richard was right to point out that Form Criticism combines the identification of a literary genre with the social setting which provides the context for understanding its meaning. It is the latter, "social setting", that is referred to normally by the German phrase Sitz im Leben (plural = Sitze im Leben).
For example, you might read: "There was a young man from Dundee..." - what kind of information is this? Is it historical? personal? Or, do I recognize that it belongs to the genre of "limerick", and so not take the information at face value at all, knowing it's just a joke? That's the value of awareness of Sitz im Leben.
In terms of biblical hermeneutics, this example has more obvious "bite" than the one Richard supplied: if we are not socialized into the "meaning" of this genre, we are at risk of wholly misunderstanding its import.
Sitz im Leben and Form Criticism
There is, however, a further nuance when it comes to Form Criticism as practised through the 20th C and applied to biblical texts. That is, Sitz im Leben refers to the hypothetical oral form from which emerged the literary type which we can now observe. This is hugely important for understanding the way this approach functioned in its pomp: roughly from the time of Hermann Gunkel (1862-1932) and into the mid- to late-twentieth century.
For example, the courtroom "form" in the prophets (e.g., Micah 6:1ff.): the sequence, identity of speakers and witnesses, etc., all were elaborated via the hypothetical social setting as reconstructed from narrative texts, especially in Samuel-Kings. Such examples could be multiplied many times, as Form Criticism was a dominant method of biblical interpretation throughout much of the 20th C. But the connection between those narratives and the prophetic text remains hypothetical, and the distance between the putative social setting of a "form" and its later literary use could be very large indeed.
Summary
For texts from the Hebrew Bible or New Testament alike, we are not "natives", either linguistically or culturally. Any Sitz im Leben, then, must be a matter of reconstruction, not intuition. And in terms of method, this "nuance" is significant, because it injects a "buffer" or gap between life setting and literary form. In more obvious cases and/or disciplined arguments, it was not a problem. But especially in some more speculative analyses -- and of these there were, again, many -- it was a potentially fatal flaw in the method.

Further Reading

Basic orientation can be found in chapter 3 of John Barton's helpful textbook, Reading the Old Testament: Method in Biblical Study (rev. edn; Westminster/John Knox, 1996). 
A standard treatment was provided by Gene Tucker, Form Criticism of the Old Testament (Fortress, 1971). Edgar McKnight has done much the same with a New Testament focus: What is Form Criticism? (Fortress, 1967).
One of the most consistent critics of the method has been Rolf Knierim: see, e.g., his "Old Testament Form Criticism Reconsidered" = Chapter 2 in Reading the Hebrew Bible for a New Millennium: Form, Concept, and Theological Perspective, ed. by W. Kim et al (A.& C. Black, 2001).

A:

Sitz Im Leben is German for "situation in life".  This is a sociology concept that tries to place certain events or items in their situations in the life of the user.  
For example, if we found the lyrics to Rock-a-bye Baby and had no knowledge of the music or situation in life where it was used, it would seem pretty harsh:

when the bough breaks, the crady will fall
  and down will come baby, cradle and all

However, the Sitz Im Leben of this song is that of a lullaby, attempting to get a child to go to sleep.  Therefore we could use that knowledge to assume that the song was sung quietly and delicately.

In form criticism, this concept is used to attempt to understand sociological setting of the genre of a particular passage in order to determine how these passages contribute to the text as a whole.
So, form criticism follows these general steps:

Identify the form or genre (parables, proverbs, hyperbole, epistle, love poem, song, etc)
Determine the "situation in life" in which this form or genre is used (Sitz Im Leben)
Find out how passages with similar genres contribute to their text and attempt to apply those concepts to the passage in question

So, Sitz Im Leben is the essential key that ties together the different passages found throughout the Bible: songs, poems, epistles, etc.

