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The spin number is the Pauli exclusion principle says you cannot have two of the same thing in the same orbital, so they must have opposing spin to counteract each other's magnetic forces. Up and down are the two different directions.
The spin magnetic quantum number is important because it describes the direction in which the electron spins. There are only two spin states that can be represented with an up or down arrow. A +1/2 is indicated with an up arrow, while -1/2 is indicated with a down arrow. The arrow pointing up can be thought as the electron spinning counterclockwise at a certain rate or clockwise at the same rate by the down arrow.
The spin number is important because it shows that there are two electrons in each orbital that are spinning in opposite directions to counteract the magnetic field. The numbers are important when filling to keep track of which electrons are in which orbitals as electrons fill in +1/2 first and then fill up each orbital again with -1/2. Thus, each orbital takes one upward spinning electron before they fill a -1/2 electron.
An important thing to note with the inclusion of the spin quantum number is that no two electrons in the same atom can have exactly the same four quantum numbers. The spins are indicated, as everyone said, by +1/2 or -1/2 since the electron can spin up or down.
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