open your saved game with any text editor and look for the line that says difficulty: around line 11, change the number to anything from 0-4 (beginner-superhuman)

here's a few examples of how difficulty comes into play:

on the geoscape:

after (14 - difficulty) months, aliens will start initiating retaliation missions proactively at the end of each month, instead of just reactively when ufos are shot down.

in dogfights, UFOs will break off after a random amount of time adjusted by difficulty, will shoot at a rate dictated by difficulty, and will generate a retaliation mission based on some percentages based on difficulty.

the funding nations will judge you more harshly.

when starting a new battlescape:

enemy unit numbers are based on the difficulty, with beginner/experienced being "low", superhuman being "high" and veteran/genius being a random number between the two.

enemy stats gain an average rate of an extra 4% per difficulty level over beginner, and on beginner, enemy armour values and the firing accuracy stat are halved.

during the battlescape game:

the likelihood of psionic attack ranges from a 25% to a 50% chance.

the likelihood of grenade attack ranges from a 20% to a 33% chance.

explosive efficacy takes difficulty into account when weighing it's decision.

alien blaster launchers are restricted to a certain number of waypoints relative to difficulty (6-14).

as a result of this enquiry, a code cleanup has occurred.