We should distinguish between the different types of anomalies that can arise in quantum field theory. An anomaly is a symmetry of the classical action that is not preserved in the quantum theory. A gauge anomaly causes a gauge symmetry to be broken leading to a violation of a Ward identity which is needed to ensure that unphysical polarization states and unphysical ghost states cancel. Gauge anomalies would cause the theory to be inconsistent and so these anomalies must cancel in any physically meaningful theory. In the standard model, these anomalies appear in individual Feynman diagrams having a triangular fermion loop with gauge bosons at the vertices. The anomalies associated with these diagrams all exactly cancel among themselves so that the theory is consistent. This cancellation seems accidental within the standard model and many people take this to be an indication that the standard model should be explained by a more fundamental theory where this cancellation occurs automatically.
The chiral anomaly is the non-conservation of a chiral current in quantum field theory that is conserved classically. This arises from triangle diagrams much like the gauge anomaly. However, the chiral anomaly does not cancel in the standard model. Since the conservation of chiral symmetry is not needed for consistency, the chiral anomaly does not cause any inconsistencies in the theory. The existence of the chiral anomaly has experimentally observable consequences: the chiral anomaly is physically required in order to explain the decay rate of the neutral pion into two photons since the triangle diagram that gives rise to the chiral anomaly also significantly contributes to the pion decay amplitude.
For more information, see the Wikipedia page on the chiral anomaly and see this previous question: Chiral anomaly and decay of the pion