Cover (Long War)
Cover provides three primary benefits: an increase in Defense, an increase in Damage Reduction (DR), and a decreased chance to be critically hit relative to units without cover. Furthermore, cover can obstruct Line of Sight, allowing you to sneak up on enemy pods undetected or avoid being targeted in combat.
In Long War, cover works much the same as it usually does, providing a bonus to Defense, but cover also gives you Damage Reduction (abbreviated "DR"). Low Cover gives +30 defense and .66 DR, while High Cover gives +45 defense as well as 1.0 DR. These represent increases from vanilla XCOM, which provided +20 and +40 defense, respectively, and granted no damage reduction.
Hunkering down doubles the base defense and damage reduction provided by cover: Low Cover gives an additional +30 Defense and +0.66 damage reduction when hunkered, while High Cover gives an additional +45 Defense and 1.0 damage reduction. The perk Fortiores Una has a similar effect with respect to damage reduction, providing an additional +0.66 DR for Low Cover and 1.0 DR for High Cover that stacks additively with the benefits provided by hunkering.
A unit that is Exposed (that is, lacks cover) or Flanked (see below) grants opponents +50% critical chance against it. Flying enemies do not provide attackers with this bonus, and some enemies (especially those who don't take cover) are Hardened, which reduces the chance they'll be critically hit by -60%.
Line of Sight
High cover has the advantage of breaking Line of Sight for all units on the same plane of elevation. This can allow XCOM operatives to close in on pods that have yet to be activated without provoking them. This can also lead to frustrating scenarios where a random tree can block a sniper from getting a shot on a flying target.
Elevation can effect line of sight: higher elevations can negate the sight-blocking properties of high cover, and lower elevations can make even low cover block line of sight. The latter is most commonly seen in forest maps; in general, two elevations of low cover is equivalent to high cover for line of sight blocking.
Units standing in cover can "peak" around it or "step out" to perform shots or throw items. Generally speaking, any unit in cover who is ordered to attack a target or use an item will automatically "step out" and temporarily move to an adjacent square before firing, if doing so is necessary to gain line of sight on a target. A step out can only be performed if the destination square is unobstructed, with some exceptions, and the step out shot doesn't count as actual movement and so won't provoke suppression or overwatch. For the purposes of line of sight, then, a unit is treated as occupying any square it can get a step out shot on, but this additional square doesn't grant additional attack range.
It's important to remember that while step out shots affect line of sight, they don't change a units sight distance until the unit actually performs the step out shot. If an unactivated pod of aliens is positioned one tile out of vision range, a unit sitting in cover can't see or target them normally, but will activate them if the unit performs a step out attack that brings the unit closer to the aliens. Similarly, units with Covering Fire who gain vision on a unit while the unit is performing a step out shot can make an overwatch attack, even if they were previously out of range.
Cover only provides it's protection in a specific direction. A unit can be Flanked by moving parallel to or past the cover, which completely removes all benefits provided by the cover for attacks made by the flanking unit as well as granting the attacker the +50% critical chance bonus for flanking the target. Note that cover continues to apply it's full defensive benefit until the unit is flanked, unless the Aiming Angles second wave option is enabled. Furthermore, a unit is granted the benefit of the best possible cover it can apply against a particular attacker, assuming said cover is not flanked relative to the attacker; a unit protected by bi-directional corner cover which is flanked relative to only one of the two directions still can apply the protection of the other direction against attackers and isn't counted as flanked, while a unit not flanked relative to either direction applies the benefit of the better of the two.
Cover can be destroyed by bullets or explosions, and any unit left without cover is Exposed. Cover is generally a little tougher to destroy than it is in vanilla due to increased health and reworked mechanics; firearms of all types take much longer to destroy cover as they receive a significant penalty to the damage they deal relative to explosives, while the damage falloff explosives suffer based on distance from the origin of the explosion, which was introduced in Long War, mean that they'll no longer reliably blow up everything highlighted in red like they tended to do in vanilla. This can be counteracted by perks such as Sapper, which makes explosives deal triple damage to all cover and destructible terrain, and Tandem Warheads, which removes the damage falloff. Either one is typically sufficient to turn explosives into highly reliable cover destruction tools.
Finally, many kinds of attacks aren't affected by cover. Anything which doesn't roll a hit chance, like explosives or Suppression, isn't affected by the bonus defense cover provides. And the bonus DR does nothing against attacks which don't roll damage, like Psi Panic or Flashbangs. If an enemy is well protected, consider alternate means of attacking them.