Q. What does this mean " The mature erythrocytes do not have a nucleus" ?.
Pretty much exactly what it says - that mature erythrocytes, or red blood cells, do not have a nucleus. It's how they get that way, and why, that is interesting.
Erythrocytes are continually produced in the red bone marrow, in a process called erythropoiesis. The cells do start out with a nucleus, as well as all the other organelles normally found in a mammalian cell. It takes about a week for each red blood cell to be produced and to mature; during that week, each cell enucleates, or ejects its own nucleus. This apparently occurs to create space inside the cell for hemoglobin, a large complex protein molecule that is involved in oxygen transport. If the cells kept the nucleus, they would likely be too large to fit through the capillaries.
Because the cells have no nucleus and thus no DNA, they cannot heal when damaged, so an erythrocyte has a functional lifespan of about 100 days. Damaged red blood cells are removed by macrophages and their parts are recycled within the body.