Quotations
(1) A particle can be in two places at once
This is very bizarre sounding, but experiments have shown that until a measurement is made, many particles can act as though they are in more than one place at once. For instance an electron being fired through a board with two small slits, if it is not measured doing so, will show an interference pattern on a screen at the back that registers impacts. This illustrates that the particle, let's say an electron, does not go through either one slit or the other, but rather it goes through both! So the particle is in a very real sense in more than one place at once. Superpositions of particles could potentially lead to super-fast quantum computing.
(2) You can never be certain where a particle is
You can never be certain of both the location and the speed of a particle at the same time according to quantum mechanics. If you measure its location precisely then lose the ability to accurately record its speed, and if you know what speed it is travelling at then you lose accuracy as to exactly where it is.
(3) The power of observation
Schrodinger's cat is a famous thought experiment illustrating the seemingly paradoxical nature of quantum theory and how observation make such a difference, almost forcing the universe to choose a particular path. In this a cat is put in a box and sealed in, with some radioactive matter that has a 50/50% chance of decaying and if it does it releases some poison to kill the cat. Until you open the box, you do not know if the cat is alive or dead, but in the quantum world (ignoring things such as decoherence) then the cat would be in a sense both alive and dead until you open the box and force the universe to choose whether the cat is alive or dead. This gives so much power to the act of conscious observation and this may either be a profound fact about reality or a confusion of some sort on our part.
(4) The measurement problem
This is the official name given to what is outlined above - the problem of how a measurement has such a profound impact on the world. In the lingo this is to do with the mechanism of wavefunction collapse: how it comes to be that the various possibilities with their assigned probabilities collapse into the single actuality at the moment of measurement/observation. It is this problem that has given rise to the various different interpretations of the theory which come in flavours such as 'don't think, just do the calculations' (Copenhagen interpretation) to the most outrageous breach of Occam's razor, but nonetheless still very popular, in the multiverse or many-worlds interpretation, where at each decision point the universe splits into each possibility in a separate universe.
(5) Virtual particles can pop into existence
You might think that empty space was just that - totally empty and a vacuum. However at small enough scales, it is now believed that something called the quantum foam exists, where spacetime is a seething mass. Sometimes particles can pop into existence, borrowing energy from the universe, and then they disapear again almost straight away. This is a strange world indeed where particles appear and annihilate back into nothingness with regularity!
(6) Black holes are not totally black
Due to the item number 5, black holes emit radiation which is known as Hawking radiation, and so eventually over an insanely huge and hard to comprehend period of time, may lose their energy and disappear. Basically the idea is that when a particle and anti-particle 'virtual' pair pop into existence, sometimes one will be the other side of the event horizon and therefore the pair cannot annihilate. Thus the black hole effectively radiates away the actual particle.
(7) Quantum Tunnelling
Standard physics / mechanics can be and is flouted by quantum mechanics! This as the name suggests allows a particle to burrow through or tunnel through something (a barrier) that it should not, according to classical theory, be able to get out of, due to the relative energies involved. This discovery was essential to understanding how things like radioactive decay work. It results from the uncertainty principle or wave/particle duality of matter.
