What happens when the kidneys stop working?

When the kidneys are unable to perform their functions, excess fluid and dregs accumulate in the blood. This results in water and salt retention, which can lead to an onset of oedema (excess of fluid in the tissues) and higher blood pressure. The excess fluid can also accumulate in the respiratory system, resulting in acute pulmonary oedema (fluid in the lungs); this condition may, in some cases, lead to death.
Other possible complications from kidney failure include:

electrolyte alterations, in particular hyperkalaemia (high potassium concentration in the blood), which may cause an onset of cardiac arrhythmia
metabolic acidosis, characterised by low blood pH and bicarbonate deficiency
anaemia, due to a lower production of erythropoietin
alterations in calcium and phosphorus metabolism, resulting in increased parathyroid hormone, bone lesions and vascular calcifications.

Kidney failure can be:
acute: the kidney functions are compromised very quickly (it can be reversible)
chronic: the deterioration of kidney function is slow, progressive and irreversible.