What types of aircraft perform military airlift operations?
There is no single "key player". It depends on the task. You need big, long-haul transporters like the C-5 or the An-124 to move large amounts of troop and equipment quickly. This is called strategic airlifting and is done with turbofan-powered aircraft which are similar to airliners. But they need prepared and secured air strips of considerable length (the NATO standard airstrip is 8000ft long).And then you need tactical transporters which can fly as close to the action as possible, and they must have STOL capabilities and defensive equipment to survive their missions. This requires mostly turboprop powered planes like the C-130, the C-160 or the An-12 for big equipment or the CASA 235 or the An-26 for smaller units or parts. The C-17 was meant to be a strategic airlifter with some tactical capabilities, but it cannot replace the smaller turboprops entirely. The A-400 and the An-70 are large tactical transports with some strategic capabilities, but this comes at the expense of usability on unprepared or improvised airstrips.
And then you need helicopters for the missions which require the highest amount of flexibility and closeness to the action.
All three depend on the other, and only together they become effective. That is why I cannot give you a key player.

Key characteristics of strategic transporters:


turbofan engines
swept wings, high flight Mach number and range
high wing, so the fuselage is close to the ground for easy loading and unloading
large nose doors and/or ramp in rear fuselage.



Key characteristics of tactical transporters:


turboprop engines (mostly)
straight wings with extensive high-lift devices for STOL capabilities
moderate flight Mach number and range
high wing, so the fuselage is close to the ground for easy loading and unloading, low loading of landing gear for soft ground capabilities
ramp in rear fuselage.



Key characteristics of helicopters


that whirly thing on top