Exemplified by the polarized states of photons, we make a concise, accurate and easily comprehensible introduction to the concepts of quantum mechanics, quantum states, quantum cryptography, quantum entanglement and quantum teleportation. First, the fundamental ideas of quantum mechanics and the meaning of quantum states are introduced through the comparison with classical physics. Then the BB84 scheme of quantum key distribution in quantum cryptography is explained. Afterwards, the concept of quantum entanglement is introduced, with the emphasis that it does not violate the theory of relativity. On the basis of this, quantum teleportation is explained, with the necessity of classical communication in the process emphasized.