“With each level of this Perk, you gain an additional 4 Hit Points every time you advance a level. This is in addition to the Hit Points you already gain per level based off of your Endurance.”— Fallout 2 In-game description
Every time the player character gains a level, they gain an additional four Hit Points for each rank of this perk. With two ranks of Lifegiver, that's +8 HP each level, which is more than 10 points in Endurance would give you.
Although it's not clear from the description, Lifegiver grants +4 HP for every level gained, including the level when the perk was taken. (Fallout Tactics does not give the HP on the level you take the perk.)
Fallout 3, Fallout: New Vegas
“With the Life Giver perk, you gain an additional 30 Hit Points.”— Fallout 3 description
This perk also gives the player character unique dialogue choices when speaking with some specific characters, like Tree Father Birch in Oasis.
In the data files, it is listed as being available in 3 ranks, and loading screens will mention multiple "levels" of the perk.
The corresponding image is a Vault Boy version of Jesus.
In Fallout: New Vegas, the health that Lifegiver grants is excluded from the 50% calculation for Ferocious Loyalty. [verified]
“You embody wellness! You also receive an instant Health increase, which has no downsides (except it takes slightly longer to inflict Nerd Rage! on a set of foes). If you need extra Health, this is a must-take perk.”— Fallout 4 Vault Dweller's Survival Guide description
You instantly gain another +20 maximum Health, and slowly regenerate lost Health.
This is best as an early-game perk. The math is such that, once one reaches Level 32, they would have been better off just putting 1 more point into Endurance than in getting a level in the Lifegiver perk. However, the benefits of +20 Health early game can be considerable.
The health regeneration gained from the third rank grants the player 0.5% health every second. However, health is not regenerated while in combat (i.e. when enemies are nearby.) The health regeneration is fast enough (out of combat) to be a real convenience.