Nephrons (the structures in the kidney that filter the blood and reabsorb/secrete molecules as necessary) atrophy and decline in number. This is thought to be partly caused by an impared oxygen supply due to the stiffening of arteries.
Glomerular filtration rate (the rate at which blood is filtered by the kidneys) declines by roughly 50% between the ages of 30 and 90 in healthy individuals. However, this is thought to have relatively little effect on life expectancy.
Glomerular filtration rate in (A) females and (B) males (source).
Age-associated increases in blood pressure, cholesterol, and inflammation increase the risk of chronic kidney disease (in which the kidneys do not filter the blood quickly enough and dialysis/kidney transplant may eventually be needed).