spin up and down

Spin up and spin down are the two basis states of a spin-½ particle once you have fixed a direction to measure along, conventionally the vertical or z axis. Up means the spin component along that axis comes out positive; down means it comes out negative. They are simply the two possible answers the chosen measurement can give, and they are written with arrows pointing up and down.

These two states form a complete basis, which means any spin state at all can be written as a combination of up and down. A spin pointing horizontally, for instance, is an equal-weight superposition of up and down; if you measure it along the vertical axis you get up half the time and down half the time. The labels up and down are not the only directions a spin can have, just the two reference states you have chosen.

The choice of which direction counts as up is entirely yours. If you instead decide to measure along a horizontal axis, you get a different pair of up and down states, and what was a clean up along the old axis becomes a fifty-fifty superposition along the new one. This freedom to pick the measurement axis, and the fact that incompatible axes give superpositions, is exactly what makes spin a vivid teaching example of quantum behaviour.