Conformal anomaly is an anomaly i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory.
A classically conformal theory is a theory which, when placed on a surface with arbitrary background metric, has an action that is invariant under rescalings of the background metric (Weyl transformations), combined with corresponding transformations of the other fields in the theory. A conformal quantum theory is one whose partition function is unchanged by rescaling the metric. The variation of the partition function with respect to the background metric is proportional to the stress tensor, and therefore the variation with respect to a conformal rescaling is proportional to the trace of the stress tensor. Therefore in the presence of a conformal anomaly the trace of the stress tensor has a non-vanishing expectation.
In string theory, conformal symmetry on the worldsheet is a local Weyl symmetry and must therefore cancel if the theory is to be consistent. The required cancellation implies that the spacetime dimensionality must be equal to the critical dimension which is either 26 in the case of bosonic string theory or 10 in the case of superstring theory. This case is called Critical String Theory. There are alternative approaches known as Subcritical String Theory in which the space-time dimensions can be less than 26 for the bosonic theory or less 10 for the superstring i.e the four dimensional case is plausible within this context.
In quantum chromodynamics in the chiral limit, the classical theory has no mass scale so there is a conformal symmetry, but this is broken by a conformal anomaly. This introduces a scale, which is the scale at which colour confinement occurs. This determines the sizes and masses of hadrons, including protons and neutrons. Hence this effect is responsible for most of the mass of ordinary matter. (In fact the quarks have non-zero masses, so the classical theory does have a mass scale. However, the masses are small so it is still nearly conformal, so there is still a conformal anomaly. The mass due to the conformal anomaly is much greater than the quark masses, so it has a much greater effect on the masses of hadrons.)