  We classify, up to isomorphism, the 2-dimensional algebras over a field K. We
focuse also on the case of characteristic 2, identifying the matrices of
GL(2,F_2) with the elements of the symmetric group S_3. The classification is
then given by the study of the orbits of this group on a 3-dimensional plane,
viewed as a Fano plane. As applications, we establish classifications of Jordan
algebras, algebras of Lie type or Hom-Associative algebras.
