The process through which blood cells are formed and developed is known as:

Hematopoiesis is the correct term for the process through which blood cells are formed and developed. This complex process occurs primarily in the bone marrow for adults and involves the differentiation of hematopoietic stem cells into various types of blood cells, including red blood cells, white blood cells, and platelets.

Understanding hematopoiesis is crucial for recognizing how blood components are produced and regulated in response to the body's needs, factoring in elements such as oxygen levels, immune response, and clotting processes. The study of hematopoiesis is fundamental to hematology, as it also relates to various disorders that can arise from abnormalities in blood cell production, such as anemia, leukemias, and other hematological conditions.

In contrast, the other terms refer to different concepts: hematemesis is the vomiting of blood, hematocytometry involves the measurement of blood cells, and hematorrhea is not a standard term in medical terminology. Together, they emphasize the specificity required in the language of hematology, illustrating why hematopoiesis is the widely accepted term for blood cell formation.