New Mechanics (LWR)
In General
If you have played the Long War mod or even just the base game (XCOM: EU/EW), many of the game mechanics in Long War Rebalance will feel familiar. However, there are some very significant changes, some of which are described below. Two of the changes you may notice off the bat is there are overall less missions (though this can be modified with Second Wave options) and the speed of the tactical game is slightly faster (can be modified by editing the dgc.ini).
Squadron Unleashed
You can now send multiple planes at the same time to engage UFOs. Also, UFOs will generally outclass what XCOM can put into the air as the game progresses so do not expect to win the air game or have complete domination of the skies. This is further described here: Air Combat.
Fatigue and Injury
Soldiers now accumulate fatigue based on how long they are in each battle. As well, you can send fatigued soldiers on missions but they have a chance to lose AP by being fatigued (or even exhausted). This is further described here: Injury and Fatigue.
Grey Market
The Grey Market no longer starts unlocked and must be unlocked through a Foundry Project. Early game credit flexibility comes more in the form of Council Loans.
Pod Activation
Enemy pods will now activate when they see an active pod or a corpse. This can lead to a chain of activations, and a more dangerous situation. This is intended and mitigated by other new game mechanics that will sometimes cancel the scamper movement of pods when they activate in this way, leaving them out in the open and more vulnerable. Sometimes they will also have their weapons start with no ammo which buys your squad a turn to prepare.
Any XCOM soldiers with 0AP remaining will gain a single move only scamper action when an alien pod is revealed: XCOM Scamper
Overwatching
Your soldiers can no longer overwatch unless they have a targetable enemy, and then their overwatch fire will only apply to those enemies that they could target when they started their overwatch. This removes the tactic of overwatch creeping from the game. While this is a big disadvantage for your squad, this rule also applies to the aliens.
Overwatching has also been buffed in a few ways. Click here for more information: Overwatch, Reaction Fire
Sneak Mode
When all active aliens cannot see any XCOM at the beginning of their turn, those aliens may enter Sneak Mode. Sneak Mode is where there aliens will actively try to move around and flank XCOM instead of running directly at them. As well, aliens that can use cover who enter Sneak Mode have a chance to become hidden.
For more information click here: Sneak Mode
Steadying
Your soldiers can no longer steady their weapons unless they have a targetable enemy. This prevents steady creeping.
Steadying has also been buffed. Click here for more information: Steady Weapon
Tactical Retreat
During most missions you can press 'Z' to call for a tactical retreat. This will drop an extraction beacon and Bigsky (the Sky Ranger pilot) will fly to your location and perform a mobile extraction within 1-3 turns.
New Abilities and Mechanics
Long War Rebalance has a lot of new features and quality of life upgrades. Many are detailed here and here.