std::forward exists because of a quirk in how && works under type deduction.
Under type deduction, the T in T&& will bind to one of 3 possibilities. If being deduced from an lvalue int&, T will bind to int&. Then int& && is just a int&. If being deduced from an lvalue int const&, T will bind to int const&, and int const& && is int const&. If being deduced from an rvalue int of some kind, T will bind to int, and int&& is int&&.
std::forward is a utility function to reverse that map. The three pertinent signatures of std::forward<> are: T& std::forward<T&>(T&) or T const& std::forward<T const&>(T const&) or T&& std::forward<T>(T&&)
All of this ends up being exceedingly useful when doing the technique known as "perfect forwarding", where you use T&&t in a type deduction context, then std::forward<T>(t) to pass on the "same type" as was deduced from to another call.
Note that there are a few simplifying lies above. There are is also the possibility of T const&& which is pretty obscure type-wise, as an example. I probably glossed over some details of how the type deduction works, and the terms rvalue and lvalue don't fully reflect the full 5-fold (or is it 6?) different kinds of variable values in C++11.