The chiral anomaly, axial or the ABJ anomaly (Adler-Bell-Jackiv anomaly) is the type of $U(1)$ global anomalies found in some quantum field theories where some currents are not conserved (they do not follow the typical formula $dJ=0$). This happens when the spinorial representations (fermions) classically exhibit a chiral symmetry (the theory is valid for left- and right-handed representations, i.e. Dirac equation for chiral massless fermions has a chiral symmetry ) nut upon quantization it is lost (because in the path integral, the measure $[D \psi]$ does not have the corresponding symmetry). Usually the anomaly factor is related to some topological invariant and is associated with tunneling and instantons (and the Chern number of the vacuum etc etc...). Another way to express it is to say that the chiral anomaly is a pathology of one-loop fermion triangle diagrams at least at one of the three vertices of the diagram.
The usual example is the chiral anomaly of the Standard Model which results in having only left-handed fermions (right-handed anti-fermions) interacting in the electro-weak processes. The same thing does not happen for QCD though because QCD is invariant under parity. Another example where the chiral anomaly shows up is the neutral pion decay to two photons.
You can find much better information at Peskin & Schroder or similar introductory, and not only, QFT books. If you do not find this answer satisfying enough I could try to throw some maths in but let me know since you can find plenty of information on the subject.