Recall that the Ising model has energy
For J>0 we have a ferromagnet. Energy is smallest when neighboring spins
are aligned, either 
or 
.
Ground state (T=0 lowest energy state) is when all spins are aligned,
or parallel (all +1 or all -1).
For J<0 we have an antiferromagnet. Energy is smallest when neighboring
spins are opposite, either 
or 
.
Ground state is when all spins are antiparallel, that is, a checkerboard
configuration for a square 2D lattice.
Thus there are only two possible ground states for Ising ferromagnets and antiferromagnets on a square lattice.