When trying to wrap your brain around renal failure, it is easiest to think about the 3 main types of renal failure: i.e. PRE-renal (problem is before the kidney), INTRA-Renal(damage to the kidney itself), and POST-Renal(obstruction after kidney).
Prerenal failure is usually caused by hypovolemia, whether that is by sepsis, severe dehydration, CHF (decreased perfusion), among others. The kidneys get ischemic and slough off the tubular lining. Here you would see symptoms within 1-2 days of the initial insult. It progresses very rapidly with rising BUN/Cr, and can be fulminant (uremic encephalopathy) unless corrective measures are taken, i.e. IV NS boluses out the ass or dialysis in rare cases.
Intrarenal Failure is probably the most complicated, as it can be caused by a myriad of things, i.e. glomuleronephritis, rhabdomyolysis, Polycystic Kidney Dz, Hypercalcemia, Hypertension, Diabetes (most common cause), Lupus, other autoimmune Dz, the list is long and distinguished. The timeline for symptoms is extremely varied, ranging from decades for DM and HTN, to hours/days for certain types of glomerulonephritis. Treatment is directed at the underlying causes.
Postrenal is the most simple. Something past or at the Uretero-Pelvic Junction is blocking the outflow of urine, thus causing elevated BUN/Cr. Treatment is to remove the obstruction. This is usually caused by prostatic hypertrophy or stones.
