(Note: my full qualifications are an on-going Coursera course on quantum computing that covers quantum teleportation lightly. FWIW, I recommend the course.)
In vague terms, the process of quantum teleportation is the following:
* You have a qubit that you want to transmit to a friend and that you don't know anything about.
* create an entangled pair of qubits (such as photons) and separate them. Keep one yourself and give another to your friend.
* Now you have two qubits; one entangled with your friends and one that you want to transmit.
* Send your two bits through a quantum circuit (the main part being a 'controlled NOT gate').
* Your bit is now entangled with the bit that is entangled with your friends bit.
* Measure each of your bits, hence 'collapsing' them. Each measurement gives you a classical bit.
* Transmit your two classical bits to your friend over any classical connection.
* Your friend then performs an operation on his or her qubit depending on what you send him or her.
* Your friend's qubit is now in the state that your original qubit was in. Your original qubit is now in a different state.
Call it teleporation if you want; it's not that unlike some sci-fi teleportation constructs of 'beaming' things places.
Why bother with this?
* It lets you send qubits to distant places that you have only classical channels to (and previously close enough contact share two entangled qubits).
* It allows more error-resistant computations: if your qubit is difficult to construct and you wish to perform a difficult operation on it, you can actually have your friend perform the difficult operation on his or her bit - prior to teleportation - and, if it succeeds, you can then proceed with the teleportation. If it doesn't succeed, your friend may try again on another entangled qubit without messing up your difficult-to-prepare qubit. After a success and teleportation, you get the same result (up to a commutator which is usually relatively small and risk-free) as performing the difficult operation on the difficult-to-prepare qubit, but with less risk of messing it up.