Assume that Alice and Bob share an entangled qubit AB. That is, Alice has one half, A, and Bob has the other half, B.
Let C denote the qubit Alice wishes to transmit to Bob.
First Alice entangles A with C.
Alice then applies a unitary operation on the qubits AC and measures the result to obtain two classical bits. In this process, the two qubits are destroyed. Bob's qubit, B, now contains information about C; however, the information is somewhat randomized. More specifically, Bob's qubit B is in one of four states uniformly chosen at random and Bob cannot obtain any information about C from his qubit.
Alice provides her two measured classical bits, which indicate which of the four states Bob possesses. Bob applies a unitary transformation which depends on the classical bits he obtains from Alice, transforming his qubit into an identical re-creation of the qubit C.
Note in this teleportation method classical bits are transferred at the speed of light, so teleportation takes time over large distances. For example if Alice and Bob are one light year apart, Bob can reconstruct C after one year.
1) Quantum Teleportation - Wikipedia