The fuel injector, though small, is a very solidly constructed fuel system component part. Fuel injectors connect to a line full of pressurised fuel, and electric impulses from the car's computer cause the valve to open many times per second, allowing fuel to flow into your engine. Mechanical damage, electronic problems and debris caught between the valve and the orifice can all cause the valve to hang open and deliver a constant stream of fuel instead of closing like it should.
Check your exhaust pipe. Any black, fuel-smelling smoke is a dead giveaway that your engine is receiving too much fuel. A leaking injector will cause your exhaust to emit a steady stream of fuel-smoke at idle, although this may diminish at higher engine speed.
Listen to your engine. If there is a consistent, single-cylinder misfire at idle and possibly higher in the RPM range, this could also point towards a leaking injector. The constant stream of fuel leaking into one cylinder throws the air/fuel ratio, this in turn makes fuel ignition for that cylinder impossible. Note that a leaking injector will only affect its host cylinder; if there are multiple or random cylinder misfires, the problem probably lies in the ignition system.
Make a note of the fuel economy. A leaking injector will have the same effect on fuel economy as a hole in your fuel tank. The problem is compounded, however: since the additional fuel going into the affected cylinder makes combustion impossible, you're wasting not only the amount of fuel leaked by the injector but also the amount that should go into powering that cylinder. Any noticeable drop in fuel economy could well point towards a leaking fuel injector.
Feel for a power reduction when driving. The same issues that affect fuel economy also affect horsepower production. Power loss might not always be extreme; a very slightly leaking injector may only alter the air-to-fuel ratio in that cylinder enough to turn on the "Check Engine" light without significantly reducing power output.
Most modern cars have the ability to self-diagnose certain problems, including a bad air-to-fuel ratio. If your "Check Engine" light comes on, take the car to your dealer or trusted mechanic to check the codes. The code may or may not come up specifically as a bad injector, as that will depend on how sophisticated your car Engine Management System is.
Do not attempt to change any part of your cars injection system unless you are trained and qualified to do so. Otherwise you can cause serious problems to your car's Engine Management System.