X-COM has five different difficulty levels that can be selected when starting a game:
On higher difficulty levels, a larger number of aliens will appear on each mission, and their stats will be higher. The Aliens' UFOs will have an higher firing rate. Also, the score-based condition to have X-COM terminated is different. There may be other effects of difficulty: For example, the USG states that on lower levels, the chance of aliens abandoning a mission ("putting their plans on hold and reevalulate") is greater if you keep shooting down their scouts.
Dynamic Difficulty Adjustment System
Quoting from Julian Gollop's  at 'Classic Games Postmortem - XCOM: UFO Defense' at Game Developers Conference 2013:
"The way this worked was very simple. Basically, if you did very well on the last three previous tactical missions and you absolutely trounced the aliens, the aliens would ramp up their own technology level and deployments more quickly. If, on the other hand, conversely you did very badly, the aliens wouldn't throw in some extra stuff. So it kinda adjusted the difficulty as you went along."
This was never implemented, however. Instead it resulted in the now infamous difficulty bug.
The total number of aliens found in each UFO or mission type are the same for Beginner and Experienced difficulty levels. On Veteran/Genius level approximately 20% - 40% more aliens will appear than on Beginner/Experienced level (depending on UFO type). On Superhuman level 50% - 100% more aliens will appear. See the "Alien Deployment" section of each UFO page for specific numbers.
Several key alien stats will increase with each difficulty level. The alien stats which change are:
|TUs, Energy, Psi Strength, Psi Skill||100%||104%||108%||112%||116%|
These percentages are modifiers to base alien stats, which can be found on the Alien Stats page. Unlike all other aliens, Zombie stats will not increase; all stats will be at "base" level regardless of difficulty level. Zombie-spawned Chryssalids will also always have base stats. In addition the USG states the aliens' "remember" the location of X-COM units longer on higher levels.
UFO Firing Rate
The UFO firing rate is decreased by an average of 3 gameseconds for each difficulty level. See Air Combat Mechanics for the formula.
If the funding nations score X-COM badly for two consecutive months, they can decide to terminate the organization. The minimum bad score trigger is affected by difficulty: