This article is from the alt.usage.english FAQ, by Mark Israel email@example.com with numerous contributions by others.
The prescriptive rule is to use "you and I" in the same contexts
as "I" (i.e., as a subject), and "you and me" in the same contexts
as "me" (i.e., as an object). In "between you and me", since "you
and me" is the object of the preposition "between", "me" is the only
correct form. But English-speakers have a tendency to regard
compounds joined with "and" as units, so that some speakers use "you
and me" exclusively, and others use "you and I" exclusively,
although such practices "have no place in modern edited prose"
(WDEU). "Between you and I" was used by Shakespeare in "The
Merchant of Venice". Since this antedates the teaching of English
grammar, it is probably *not* "hypercorrection". (This is mentioned
merely to caution against the hypercorrection theory, not to defend
the phrase.) Shakespeare also used "between you and me".