How does induction ice form in and affect jet engines?
Induction ice can be a problem for jet engines. Other answers have explained how the ice forms. The air expands and cools in the inlet before entering the engine. The first stages of the compressor do not compress the air enough yet to warm it enough to prevent ice from forming. Low power settings can also make it easier for ice to form. Certain atmospheric conditions make ice more likely to form.

The FAA released AC 20-147, which "provides guidance and acceptable methods, but not the only methods, for demonstrating compliance with the applicable engine induction system icing and engine ice ingestion requirements." This describes different types of icing, and the effects it can have on engines. It also describes how engines should be tested to conform to the regulations for these conditions. Ice can form in various places:


The fan or compressor blades themselves
Vanes
Sensors
Bleed ducts


This ice can cause various problems:


The ice can come off and damage engine components downstream
The ice will reduce the effectiveness of blades, making it harder for the engine to compress the air
The ice can impede the flow of air in the engine
The ice can block sensors, preventing the control system from correctly monitoring the engine, which can lead to power loss or instability


One form of icing that has been an issue lately is ice crystal icing. This is caused by small ice crystals in the upper atmosphere, particularly above convective clouds. The FAA has issued an AD (Airworthiness Directive) recently about this.


  This type of icing does not appear on radar due to its low reflectivity, and neither the airplane ice detector nor visual indications reliably indicate the presence of icing conditions. Therefore, it is often undetected by the flight crew. Flight in these conditions can cause ice crystals to accumulate in the core gas flow path of the engine.


There were previous events on GE CF6-80 engines (used on 747, 767, A310, A330, and others) that the FAA looked into.


  In the events leading to those prior ADs, the ice has shed during throttle transients and in the descent phase of flight, causing temporary thrust loss.


However, there have been incidents with the newer GEnx engines on 747-8 and 787 aircraft which have been more serious.


  The new events that prompted this AD, however, have occurred during the cruise phase of flight and caused permanent damage (beyond maintenance manual limits) to the engine compressor. In all thrust loss events, data indicate that ice crystals entered and collected in the initial stages of the compressor. Engine temperature data indicate small ice accretions were shed through the core of the engine.


The engine manufacturers are working to resolve the issues on the aircraft currently affected, and in the meantime flight crews are advised to avoid weather conditions where ice crystals tend to form.


  More definitively, the AD includes immediate mitigating procedures for the affected aircraft which prohibit flight within 50nm of amber and red radar returns that are displayed below the aircraft’s flight path during operations at or above 30,000 ft, when approaching, or in, IMC or visible moisture.
  
  Note: Boeing has also published an updated Flight Crew Operations Manual Bulletin with specific operating procedures for flight crews.