In an airliner, such as an A380, what indications do the pilots have that an engine has thrown a fan blade?
Will that always result in destruction (of the engine)?
Turbofan engines are designed and tested to contain a fan blade failure. However, this is a violent event and will cause serious damage to the engine. The engine and surrounding components are subjected to large forces and vibrations, and the engine can ingest debris from the damage to the fan, causing further damage. There will likely be an investigation, and the engine will probably be scrapped due to the widespread damage to different components.
To the pilots, it will seem like a bird strike, but probably worse. There will be a loud noise as the fan blade separates and collides with the fan case and following blades. The fan will then be out of balance, so heavy vibrations will follow as the shaft loses its rotational energy. The pilots will be able to tell which engine has failed by the loss of thrust, as well as the indications of vibration and failure on the instruments for that engine.
If the fan does not get stuck, it will continue to windmill, which will result in ongoing vibration from that engine.
engineer and pilot here - formerly of JPL. I've seen blade loss tests and any modern high performance engine will violently destroy itself if a blade is lost. You can see tests of this on youtube for example at that link or by searching on "jet engine blade-out test" (without the quotes). (User fooot gave a longer example video as well.) In these tests they typically use a (very small, prebalanced) explosive charge to separate one blade from the spindle and then the engine vibrates itself to pieces.
Modern engines are constructed near the very limits of the materials used. In fact there is a special joint venture company in the US that makes a lot of the fan blades and which exists just for this purpose. They use special tools to X-ray every blade and measure the center of gravity as the slightest bubble in composites or weight anomaly would be enough to destroy an engine. When a blade is lost, the off center mass causes an engine to vibrate itself violently to death.
In person you can't really hear anything but the vibration and destruction at these tests - I couldn't even hear the explosive charge when it went off (although it seems audible in some of those videos). As fooot noted above, the pilots are informed through the usual engine-out detection methods such as loss of telemetry (and it may start a fire which would be noted and dealt with by fire suppression systems on a larger jet).
Many companies including, I know, Rolls-Royce will invite the public to fan blade out tests to show how their engine bodies control the resulting carnage. It might be a fun afternoon if you're the type to visit aviation stackexchange :) They post videos online for the same reason and you can see a lot of very nice slow-motion videos on the subject.
If you lose a "large" blade like a fan blade, or a "heavy" blade like most turbine blades, the engine vibration monitors will instantly go off the scale and the engine control system should shut the engine down automatically. To give some idea of the forces involved, losing a fan blade from a large turbofan will create an force of about 100 tons on the engine, oscillating at a few thousand RPM until the rotor slows down. In a worst case scenario, the aftermath is enough to shake the entire aircraft to the level where walking about in the passenger cabin is difficult. But aircraft have survived that and landed safely, and for long distance flights over water the plane is certified to survive the consequences for up to 3 hours flight time before reaching the nearest land.
On the other hand, if the tip of 10mm long high pressure compressor blade cracks off in the engine core, you might not even know about it until the next time the engine is stripped down for a full inspection.
Provided the engine survived the event, we are looking at all the engine parameters from the cockpit displays. Those indications are different depending on engine manufacturer. N1, N2, EGT, Oil pressure, ITT, engine vibrations indicator (VIB) and source, EPR (...). Just by looking at the numbers you cannot tell for sure if a blade separated. And it all depends also from the compressor stage the blade separated (first, last...) or if it was a fan blade.