In what is referred to as 'quantum teleportation' in the physics community, the quantum state $|\psi\rangle$ of a quantum object is transferred onto another (identical) quantum system.
Typically, the two ends of the communication channel are given names: Alice is the sender and Bob is the receiver. The most fundamental quantum system is called a qubit in quantum information theory. It is a two-state system and you can think of it as an electron spin which can be either up or down (though, there are different physical realizations of qubits). Every more complex system can be broken down into qubits, so we typically explain teleportation only in terms of qubits.
Now, Alice has a qubit in a state $|\psi\rangle$. Also, Alice and Bob have two qubits which are in an entangled state, i.e. their state cannot be described independently, but as a collective. (People say 'Alice and Bob share entanglement'.) Now, Alice can perform a quantum measurement on her two qubits which will entangle them and let Bob's qubit end up in one of four states: $|\psi\rangle, \sigma_x|\psi\rangle, \sigma_y|\psi\rangle$ or $\sigma_z|\psi\rangle$. Bob can than retrieve $|\psi\rangle$ by applying a suitable transformation on his qubit. The point is, that he doesn't know which of the four states he got, unless Alice tells him the outcome of her measurement. This is were the transfer of (classical) information comes in and it is what save teleportation from violating relativistic signal locality (i.e. no information can be transferred faster than light).
To address your original question: in quantum teleportation of atoms, they are not disassembled of destroyed. The atoms that composed captain Kirk would still be on the transporter platform after beaming (a pile of carbon atoms). Also, you would have to have the right amount and kinds of atoms in the target location. Quantum teleportation does not transport matter.
As to the copying issue: Teleportation does not duplicate the quantum state. There is even a so-called 'no cloning theorem' which explicitly prohibits a quantum state from being copied, so any transporter accidents á la The enemy within are not possible (or if you are more the TNG kinda guy there's also Second Chances).
Disclaimer: I just assumed you know that real quantum teleportation of persons is currently beyond our technological reach and will be for a long time to come. There have been experiments where single atoms and even small gas clouds of atoms have successfully been teleported but I wouldn't get my hopes up...