What are the differences between bound entanglement and free entanglement? It is shown that the non-positive partial transposition operation may be inefficient in detecting some entangled states in some quantum systems like qutrits. Hence obtaining a suitable map is useful to show that the system is entangled even though the eigenvalues of its partial transpose are positive. These states are said "bound entangled", which are different from "free entangled states". 
I wish to know what are the meanings of, and the differences between, the followings: free entanglement, bound entanglement, distillable entanglement? 
What is the correlation between the above terms? 
 A: Your definitions are already somewhat off. A state is called bound entangled if it is entangled and no pure entanglement can be distilled from it. This means that given any number of copies of the state, you cannot use LOCC operations to obtain a maximally entangled state from them. 
I don't really use the term free entangled states, but from a bunch of papers I guess these form the complement of the set of bound entangled states in the entangled states, hence those are exactly the states that can be distilled. The distillable entanglement is the amount of pure (maximal) entanglement that can be created from this state in the limit of infinitely many copies. Distillable entanglement is therefore a common measure of entanglement, which however is quite hard to calculate for mixed states. By definition of the term "bound entanglement", the distillable entanglement of bound entangled states is zero, while it is nonzero for free entangled states.
Note: Bound entangled states for qubit-qubit and qubit-qutrit systems are equivalent to states where the partial transpose is positive. This is not necessarily true in higher dimensions: While it is true that all states with positive partial transpose are bound entangled, the converse remains unknown and known as the NPT-bound entanglement problem. 
