Life on Hit (a.k.a. Life per Hit) is an item attribute introduced in Diablo III, which works much like Life Steal in that upon inflicting damage upon monsters, the nephalem hero heals Life.
In-game
Unlike Life Steal, Life per Hit heals a fixed amount of Life for every damaging attack that hits an enemy target, regardless of the Proc Coefficient and the amount of enemies hit. Note that unlike most 'on hit' effects, Life per Hit requires that you actually hit an enemy, not an object, and deal any damage with that attack. This normally does not trigger off attacks from pets and minions, attacks that deal no damage and skills that do not interrupt normal casting, attacking and channeling (i.e. do not animate, like Frost Nova or Overpower). Channeled attacks heal the character every attack period (that is, with 2 attacks per second, it will heal the player every 0.5 seconds). Damage over time attacks (like Haunt or Blizzard) only heal the caster once, whenever the first enemy is initially affected, and do not heal the caster with continuous damage.
Life per Hit can roll on rings, gloves, weapons, helms, amulets, and bracers. At level 70, maximum Life per Hit on a single item can be as high as 18429 (follower item, amulet or one-handed weapon), 27644 (two-handed weapon) or 9214 (any other item), to a maximum of 92143 from all items (119750 with Ancient Legendary bonus). Some skills, like Shadow Power, also increase this stat. Bottomless Potion of the Leech adds another 15000-20000 for 5 seconds.
A hero wishing to improve this can spend up to 50 Paragon points on the statistic. It is located in the Utility tab, and it grows in power to match his/her level, capping at 160.9 per point (8045 total) for level 70 Reaper of Souls players. An Amethyst in a weapon socket also grants up to 28000 LoH. Leech passive skill also adds up to 18507 Life per Hit (scales with level). Leech Curse will effectively grant Life per Hit equal to 2% of maximum Life against afflicted enemies.
Note that the same item may never roll both Life on Hit and Life per Kill, even if enchanted.
Development
Originally, Life per Hit healing was based on the Proc Coefficient and the amount of enemies hit. As of Patch 2.1, developers decided to rework the LoH, making it more stable and reliable. Existing LoH bonuses, including Paragon level bonus, were boosted greatly.