Kidney function is best described by the Glomerular Filtration Rate
(GFR), which describes the volume of plasma ultrafiltrate produced by the kidneys
each unit time (ml/min).
Since this is difficult to measure directly, it is estimated by the clearance of a substance such as inulin,creatinine, or urea from the blood. Inulin is difficult to measure properly, and creatinine is a good proxy.
Creatinine is a harmless substance produced as the byproduct of the degradation of creatine. It is present in all people, though it will be higher in those with more muscle activity (and thus muscle breakdown), particularly younger and more active people. It would be spuriously high in people who take exogenous creatine, and this should be taken into account.
Creatinine is freely filtered by the kidney. That is, the kidneys do not actively retain or actively excrete creatinine - they allow the amount present in the plasma before filtration to remain in the filtrate.
This is slightly simplistic: a small amount is actually actively excreted by the proximal tubule. Since Urea is resorbed, it can be slightly more accurate to average the results obtained from creatinine with those from urea; in practice, creatinine alone gives good results.
The Cockcroft-Gault Formula is used to calculate the Glomerular Filtration Rate based on serum creatinine. It has assumptions built into it about creatinine production, based on the weight, age, and sex of the patient that are not always correct, but give a good heuristic.
For a more accurate calculation, the serum creatinine concentration should be compared to the urine creatinine excretion rate. The difficulty with such a measurement is that a urine concentration is insufficient unless the amount of urine produced per unit time is measured. A 24 hour urine collection would be the gold standard. For most purposes, the calculated creatinine clearance based on the Cockcroft-Gault Formula is the easiest and best test of renal function.