The Pauli spin matrices represent the three (x,y,z) spin operators acting on a spin 1/2 particle (such as an electron): that is to say that the spin operator can be written as S = (1/2)(h-bar)σ. They are conventionally written as:

σ1 = (0  1)
     (1  0)

σ2 = (0 -i)
     (i  0)

σ3 = (1  0)
     (0 -1)

They are each unitary, hermitian and trace-free (indeed, the form a basis of sl₂(C)). Also, they anticommute in pairs, and satisfy the commutation relation [ σ_i,σ_j ] = 2iε_ijkσ_k.