Pretty much every electro optical system that doesn't have to be visible. IR LED's are cheaper than visible ones and nicely match the sensitivity of common detectors, so IR is usually first the choice.
Examples (cribbing some from above to keep the list complete):
Remote controls
Opto isolaters (isolates two sections of a circuit by optically coupling a signal from one section to the other)
Position sensors (industrial, inside VCR's, etc.)
Beam break detectors (industrial control, safety interlocks, security systems)
Optical scanners (grading "fill in the dot" sheets, etc.)
Fiber optic communications (short range; long range uses semi-conductor lasers)
And, since semiconductor lasers are specialized LED's, you can add long range fiber optic communications, pump lasers for other colors, etc.
And, as pointed out above, they can also be used to hide illumination from people, either for covertness or to avoid distraction, when used for lighting or signalling.
Cheers,
Bob