Fundamental particles such as quarks and leptons can have a spin either up or down. These spins are (obviously) opposite of each other. But what differentiates them? Let's say you examine a pair of electrons and you find out they're opposite (one up and one down). How do you know which one is up and which one is down? Do all up-spinning-particles in the universe point to an established direction? If not, how can we know the difference between an up and a down spin?
There's no universal notion of "up" or "down", just like there's no universal notion of "left" or "right" in the universe.
Speaking of "spin up" and "spin down" involves two completely arbitrary choices - a choice of axis and then a choice of "up" and "down" along that axis. You can swap the meanings of "up" and "down", and you can choose another axis to speak of "up" and "down" along. It's just a choice.
As to how to distinguish "up" and "down": For example, if you have a constant magnetic field in a direction and put an electron into it, the electron states with spin "up" along the axis of that field will have a different energy value from the states with spin "down", since the magnetic field essentially couples to the spin "direction" through the magnetic moment. Which one of these you call "up" or "down" is still somewhat arbitrary, but you can distinguish them.
Let's say you examine a pair of electrons and you find out they're opposite (one up and one down). How do you know which one is up and which one is down?
There is no way of knowing, or more accurately, there is no answer to your question. The wave function of the state you describe is a super position of the two options. There is not even any way of distinguishing between the two (they don't have individual names).