In a theory with two chiral fields , ψ1 and ψ2 with a global symmetry relating the relative phases of both fields, but at low temperatures, the correlation function 
is nonzero, then we say a fermion condensate (also called chiral condensate) has formed. The VEV above is the order parameter. See order-disorder. For example, in QCD, there is an approximate (so there's no real spontaneous symmetry breaking; the VEV will always be aligned in a fixed direction) axial symmetry which is broken because the quarks form a chiral condensate because of a pool of instantons. See Technicolor (physics) for another example. BCS in superconductivity is another.
Heuristically, what happens is a pair of fermion can form a bound state, like a Cooper pair or a meson. Then, the bound states themselves form a condensate. A Cooper pair is not electrically neutral and so, a Cooper pair condensate would necessarily break the electromagnetic gauge symmetry. Similarly, a meson breaks chirality. A phenomological description of the (composite) meson field is given by the chiral model.
To get a "feel" for chiral condensates, a good toy model to start with is the Schwinger model.
See also top quark condensate