Turbochargers are exhaust-driven compressors that shove air into an engine -- the extra air allows the engine to burn more fuel and make more power. Turbos are subject to a great number of stresses; for example, the turbo bearings are in constant contact with 420C exhaust gases. The oil seal must contend with extreme vacuum from the compressor inlet and pressure from the exhaust and the compressor blades must hold together at speeds approaching 100,000 rpm. Turbos can fail in a number of ways, some easier to troubleshoot than others.
Examine the exhaust pipe for plumes of blue smoke when you rev the engine. Oil seals are often the first thing to go, and when they do, they'll allow oil from the bearings to leak into the intake tract. The oil atomises in the compressor blades and burns inside the engine to produce blue smoke. If you see oil smoke coming from your exhaust, disconnect the turbo outlet tube and run a finger around inside the tube to check for oil residue on the walls. Consider running a high-mileage engine oil with a seal conditioner. A seal conditioner will cause your oil seal to swell slightly and seal against the shaft, buying you some time before seal replacement becomes mandatory.
Check your turbo boost pressure. If your car has a turbo boost gauge on the dashboard, than all you need to do is to watch it and check the reading against your owner's manual. If your car doesn't have a gauge, you'll need to connect a vacuum-boost gauge to one of the rubber vacuum lines on your throttle body. A low boost reading could indicate a bad turbo, but it could just as easily indicate a leak in the ducting or a malfunctioning waste gate, or turbo-control valve. Boost that quickly rises when you apply throttle and then falls off almost always indicates a leak in the ducting, waste gate sensor line or a sticking waste-gate valve.
Remove the ducting that runs from your air cleaner to the turbo inlet to inspect the compressor wheel. Inspect the wheel blades for damage and spin the wheel with a finger. The wheel should spin freely in the housing and remain spinning for a few seconds. Any resistance to movement indicates worn bearings and possibly damage to the oil seal. Your turbocharger may or may not be salvageable -- many incorporate pressed-in components that can only be removed or installed with a machine shop press. Having your turbocharger rebuilt may well cost as much as purchasing a re-manufactured unit.