AE1 was discovered following SDS-PAGEgel electrophoresis of erythrocyte cell membrane. The large 'third' band on the electrophoresis gel represented AE1, which was thus initially termed 'Band 3'.
The chloride-bicarbonate exchanger in the red cell membrane is not a pump, which would use metabolic energy. Nor is it strictly an enzyme. It is protein counter-transporter, known as band III. 
AE1 in Red Cells
AE1 is an important structural component of the erythrocyte cell membrane, making up to 25% of the cell membrane surface, indeed each red cell contains approximately one million copies of AE1.
Here it performs two functions:
Electroneutral chloride and bicarbonate exchange across the plasma membrane on a one-for-one basis.This is crucial for CO2 uptake by the red cell and conversion (by hydration catalysed by carbonic anhydrase) into a proton and a bicarbonate ion. The bicarbonate is then extruded from the cell by the band 3 molecule.
Physical linkage of the plasma membrane to the underlying membrane skeleton (via binding with ankyrin and protein 4.2). This appears to be to prevent membrane surface loss, rather than being to do with membrane skeleton assembly.
Mutations of erythroid AE1 affecting the extracellular domains of the molecule may cause alterations in the individual's blood group, as band 3 determines the Diego blood group.
More importantly erythroid AE1 mutations cause between 15-25% of cases of Hereditary spherocytosis (a disorder associated with progressive red cell membrane loss), and also cause the hereditary conditions of Hereditary stomatocytosis  and Southeast Asian Ovalocytosis 
AE1 in Alpha-Intercalated cells
A different isoform of AE1, known as kAE1 (which is 65 amino acids shorter than erythroid AE1) is found in the basolateral surface of the alpha-intercalated cell in the cortical collecting duct of the kidney.
This is the principal acid secreting cell of the kidney, which generates hydrogen ions and bicarbonate ions from carbon dioxide and water-a reaction catalysed by Carbonic anhydrase.
kAE1 exchanges bicarbonate for chloride on the basolateral surface, essentially returning bicarbonate to the blood.
Mutations of kidney AE1 cause distal (type1) renal tubular acidosis, which is an inability to acidify the urine, even if the blood is too acid.
These mutations are disease causing as they cause mistargetting of the mutant band 3 proteins so that they are retained within the cell or occasionally addressed to the wrong (ie apical) surface.
^ Hunter M (1977). "Human erythrocyte anion permeabilities measured under conditions of net charge transfer.". J Physiol268 (1): 35-49. PMID 874904.
^ LJ Bruce (Nov 2005). "Monovalent cation leaks in human red cells caused by single amino-acid substitutions in the transport domain of the band 3 chloride-bicarbonate exchanger, AE1.". Nature Genetics37 (11): 1258-63.
^ Jarolim p (1991). "Deletion in Erythrocyte Band 3 Gene in Malaria-Resistant Southeast Asian Ovalocytosis". PNAS88: 11022-11026..
Tanner MJ (1993). "Molecular and cellular biology of the erythrocyte anion exchanger (AE1).". Semin. Hematol.30 (1): 34-57. PMID 8434259.
Chambers EJ, Askin D, Bloomberg GB, et al. (1998). "Studies on the structure of a transmembrane region and a cytoplasmic loop of the human red cell anion exchanger (band 3, AE1).". Biochem. Soc. Trans.26 (3): 516-20. PMID 9765907.
Inaba M (2002). "[Band 3: expanding knowledge on its functions]". Seikagaku73 (12): 1431-5. PMID 11831035.
Tanner MJ (2002). "Band 3 anion exchanger and its involvement in erythrocyte and kidney disorders.". Curr. Opin. Hematol.9 (2): 133-9. PMID 11844997.
Shayakul C, Alper SL (2004). "Defects in processing and trafficking of the AE1 Cl-/HCO3- exchanger associated with inherited distal renal tubular acidosis.". Clin. Exp. Nephrol.8 (1): 1-11. doi:10.1007/s10157-003-0271-x. PMID 15067510.