The short answer: Each arrow represents an electron in the atom. The direction of the arrow represents the spin the electron has.
A longer answer:
Several experimental observations can be explained by treating the electron
as though it were spinning. The spin can be clockwise or counterclockwise,
and so there are two possible values of the 'spin quantum number' that describe the electron.
The quantum theory was able to explain the experimental results if the spin
quantum number was taken to be either +1/2 or -1/2. An up arrow on an orbital
diagram means one of these values, and a down arrow means the other (try not to worry about which is which. It really doesn't matter.)
Why should you care which way the electrons are spinning?
Spin determines how many electrons can occupy an orbital, and also, which orbitals electrons will fill first within a subshell.
Each electron in an atom is uniquely labelled by its set of 4 quantum numbers.
(This is the Pauli Exclusion Principle, for which Wolfgang Pauli received a Nobel Prize
Three quantum numbers (n, l, and ml) are needed to describe the orbital the electron occupies.
Since there are two possible values for the fourth quantum number,
you may place no more than two electrons in an orbital when filling in the orbital diagram. You must also ensure that when there are two electrons in the same
orbital, they have opposite spins. Electrons repel each other, and tend to align their spins whenever possible. When filling up a subshell on an orbital diagram, then, you must spread the electrons among different orbitals to keep them as
far apart as possible, and line up their spins. (This is called Hund's rule).
The discovery that electrons and other particles have a 'spin' property had
profound consequences in physics and chemistry.
Technological applications include magnetic resonance imaging (MRI),
which allows detailed mapping of soft tissues in the body, mainly for diagnosis and
monitoring of diseases like cancer. Future applications will include 'quantum
computers' which will exploit spin states to store information on an atomic size scale.
Author: Fred Senese firstname.lastname@example.org