Shinobi Stats

Ninjutsu

In-game description: "Makes hand signs faster, so jutsu are cast faster."

Hand-sign speed: each sign normally takes 0.5 seconds. Your speed multiplier is 1 + 0.04 per Ninjutsu point, up to 3x (reached at 50). At Ninjutsu 25 signs are twice as fast; at 50 they are three times as fast.
A full sign sequence takes a 0.2-second wind-up, then the signs, then a 0.25-second wind-down.
A jutsu's details show this as "Hand-sign speed xN (Ninjutsu)".
Chakra regeneration: +8 chakra per second per point, on top of the base 25 per second. At 50 Ninjutsu you regenerate 425 chakra per second.
Jutsu damage: your jutsu deal 1 + 0.006 x Ninjutsu times their damage (+0.6% per point), up to x1.3 (reached at 50).
Requirement: Ninjutsu is the most common stat requirement on jutsu. About 200 skill-tree nodes ask for some.

Taijutsu

In-game description: "Increases unarmed (no weapon) M1 damage and the damage of taijutsu moves."

Taijutsu moves: Leaf Whirlwind, Leaf Hurricane, Dynamic Entry and Primary Lotus deal +2% per point of Taijutsu: their listed damage at 0, twice it at 50.

Bare-handed M1 damage: a full punch deals Taijutsu x 0.5 damage, never less than 1 (a normal fist). At the cap of 50 that is 25 damage per full-strength blow (the mod's health scale runs to 500; see Health and Damage Scale).
"Bare-handed" means your main hand holds nothing that adds attack damage. Holding a release, a block, food or any other non-weapon counts as a fist.
The attack cooldown and critical hits still scale it: a half-charged swing does half.
Some techniques add a flat amount to every bare-handed blow before the multipliers: each open gate of the Eight Gates adds 6, Unrivaled Strength (Boil Release) adds 15 and Kagutsuchi's black-flame armour adds 8.
Being Expanded (the Akimichi-style Expansion effect) makes it x1.5, and Butterfly Mode makes it x2. A jinchuriki's form can multiply it further, and stances and buffs (Gentle Fist and so on) multiply it after that. A full armor set with a taijutsu bonus adds to it too.
Requirement: Taijutsu-type techniques ask for it (24 skill-tree nodes).

Genjutsu

In-game description: "Required to use genjutsu techniques."

Genjutsu is a requirement stat: genjutsu techniques ask for Genjutsu points. There are 8 skill-tree nodes that need it: Hell Viewing (2), Temple of Nirvana (4), Genjutsu Release (5), Mind Transfer Projection (8), False Surroundings (10), Frog Song (10), Mind Disturbance (14) and Mirror Heaven and Earth (18).

Kenjutsu

In-game description: "Increases weapon damage."

Kenjutsu powers the Kunai's melee hit: a kunai swing deals the greater of 2.5 or Kenjutsu x 0.5 (25 at Kenjutsu 50). That number is already on the mod's damage scale, so it is not multiplied again. Swing charge and critical hits scale it as usual.
Kenjutsu also adds to every sword and blade (the Katana, any sword, a ninjato, Samehada...): a full swing deals the weapon's own damage plus Kenjutsu x 0.5 (+25 at Kenjutsu 50), scaled by swing charge and critical hits.
Requirement: Kenjutsu-tree techniques ask for it, such as Silent Killing (3), Chakra Flow Blade (6), Crescent Moon Dance (8) and Hidden Mist Blade (16).

Medical

In-game description: "Raises max health (up to 500 at the cap), and is required for healing jutsu."

Maximum health: max health = 20 + 480 x (Medical / 50), with the fraction never above 1. Each point adds 9.6 max health and 50 Medical gives 500 max health, the most Medical can give. Clan bonuses count, but they can never push the Medical part past 500.
Armor adds on top: max health from armor pieces is added after that and isn't limited by the 500, so good armor can take you past 500 max health.
You respawn at full (boosted) health.
Medical Release: the Medical tree's jutsu need Medical Release first (see the Medical Tree page). The Medical stat still raises max health without it.
Requirement: medical ninjutsu ask for it: Mystical Palm (3), Bite Heal (6), Chakra Scalpel (8), Poison Mist (9), Delicate Extraction (11), Massive Regeneration Aura (15) and Creation Rebirth (25).
Medical also strengthens healing jutsu (see Healing and Medical Ninjas).

Intelligence

In-game description: "Raises the chance of identifying Unidentified DNA (DNA page). It does nothing else."

Skill Points can't be spent on Intelligence: its + button on the Shinobi Build page is always greyed out. Your Intelligence comes only from your clan's bonus (for example the Nara, Aburame, Jiryoku, Kami-E and Sumie clans give +1).
Intelligence affects identifying Unidentified DNA on the DNA page. Every point raises your success chance, up to a cap that is reached at 28 Intelligence; points past 28 add nothing more.
If identification fails, the sample is destroyed.