Here we report a quasi-one-dimensional S=1 chain compound NiTe2O5. From the comprehensive study of the structure and magnetic properties on high-quality single-crystalline NiTe2O5, it is revealed that NiTe2O5 undergoes a transition into an intriguing long-range antiferromagnetic order at TN=30.5K, in which longitudinal magnetic moments along the chain direction are ferromagnetically ordered, while their transverse components have an alternating ferromagnetic-antiferromagnetic coupling. Even though the temperature dependence of magnetic susceptibility represents an archetypal anisotropic antiferromagnetic order, we found that critical behavior of unconventional nature with α′∼0.25 and β∼0.18 is accompanied by the temperature evolution of the antiferromagnetic order parameter.