What is the relationship between tail recursion with other recursions?
We need to distinguish two things:


Tail-calls are calls to functions that are made in tail position. Tail-recursion is a recursive tail-call.
Tail-call optimization is the process by which tail-calls are turned into loops or GOTOs in compilers.


The second, as others have stated, is an implementation detail that falls in the theory of compilers, and isn't really recursion theory.

However, it is absolutely possible to study tail calls in the context of computability. In particular, there is something called Continuation Passing Style.

There exist CPS-transformations which can turn any lambda-function into one in continuation passing style. So the set of languages computed using only tail-calls is equivalent to the set of recursively-enumerable languages.

CPS has many applications in the theory of programming languages, and the concept of tail recursion is useful for more than just the tail-call optimization.