Purine degradation takes place mainly in the liver of humans and requires an assortment of enzymes to degrade purines to uric acid.  First, the nucleotide will lose its phosphate through 5'-nucleotidase. The nucleoside, adenosine, is then deaminated and hydrolyzed to form hypoxanthine via adenosine deaminase and nucleosidase respectively. Hypoxanthine is then oxidized to form xanthine and then uric acid through the action of xanthine oxidase. The other purine nucleoside, guanosine, is cleaved to form guanine.  Guanine is then deaminated via guanine deaminase to form xanthine which is then converted to uric acid.  Oxygen is the final electron acceptor in the degradation of both purines. Uric acid is then excreted from the body in different forms depending on the animal.
Free purine and pyrimidine bases that are released into the cell are typically transported intercellularly across membranes and salvaged to create more nucleotides via nucleotide salvage. 
For example, adenine + PRPP --> AMP + PPi.  This reaction requires the enzyme adenine phosphoribosyltransferase.  Free guanine is salvaged in the same way except it requires hypoxanthine-guanine phosphoribosyltransferase.
Defects in purine catabolism can result in a variety of diseases including gout, which stems from an accumulation of uric acid crystals in various joints, and adenosine deaminase deficiency, which causes immunodeficiency.