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The Pauli Matrices

The Pauli Matrices are a set of 2×2 matrices which are indispensible in quantum mechanics. They are as follows:

σ1 = (
0 1
1 0
)
σ2 = (
0-i
i 0
)
σ3 = (
1 0
0-1
)

They may also be expressed as components of the vector σ as follows:

σ = (σ1, σ2, σ3)

You might also see them referred to as σx, σy and σz. These are exactly the same as σ1, σ2 and σ3, it's simply a different convention for labelling the different components of the vector σ.

Algebraic Properties

Warning: This section is not written in Plain English!

The Pauli matrices are hermitian and unitary.
They have the following commutation and anti-commutation relations:

[σi, σj]=2iεijk
{σi, σj}=2δij

Putting these relations together gives us an equation of what happens when we multiply any two Pauli matrices together:

σi σj  =  iεijk + δij

QWait! I don't quite see where you got this equation from!
AI took the commutation [  ,  ] relation, the anti-commutation relation {  ,  }, added them together then divided by 2.
Why? Because in general for two symbols A and B; commutation means  [A, B] = AB - BA,
and the anti-commutaion means: {A, B} = AB + BA.
So [A, B] + {A, B} = AB - BA + AB + BA = 2AB
Dividing this by 2 gives AB.

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