Wikipedia
A dredge-up is a period in the evolution of a star where a surface convection zone extends down to the layers where material has undergone nuclear fusion. As a result, the fusion products are mixed into the outer layers of the stellar atmosphere where they can appear in the spectrum of the star.
The first dredge-up occurs when a main-sequence star enters the red-giant branch. As a result of the convective mixing, the outer atmosphere will display the spectral signature of hydrogen fusion: the C/ C and C/ N ratios are lowered, and the surface abundances of lithium and beryllium may be reduced.
The second dredge-up occurs in stars with 4–8 solar masses. When helium fusion comes to an end at the core, convection mixes the products of the CNO cycle. This second dredge-up results in an increase in the surface abundance of He and N, whereas the amount of C and O decreases.
The third dredge-up occurs after a star enters the asymptotic giant branch and a flash occurs along a helium burning shell. This dredge-up causes helium, carbon and the s-process products to be brought to the surface. The result is an increase in the abundance of carbon relative to oxygen, which can create a carbon star.
The names of the dredge-ups are set by the evolutionary and structural state of the star in which each occurs, not by the sequence experienced by the star. As a result, lower-mass stars experience the first and third dredge-ups in their evolution but not the second.