There are 44 passive talents (11 of which are exclusive to the Summoner) and 5 possible slots to equip them.
Passive talents max out at level 4, 5, or 10.

1. You start off with the first one
   
2. Beat Rog's Mode in F-Rank
   
3. Complete the D-Rank Decent into Hell Challenge (Then buy the upgrade in the shop)
   
4. Beat Rog's Mode in C-Rank
   
5. Beat Rog's Mode in B-Rank

Talents will be ranked from one to five stars for their scalability in the late game.
(Potentially biased as all these are based on my playstyle, so if you feel like something should be rated higher or lower, please comment below.)

To level up talents, it’s quite simple: Play games. To level them up faster, you would want to do higher ranked games, as well as increasing the starting corruption modifier.

Max out starting corruption and play the Glass bone challenge at Level 4.
You can find this in the Survivors Mode F-Rank challenges.

Talents that directly or indirectly increase damage.

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Increase Damage by 4% for every meter between you and the damaged enemy.
*How to obtain: Kill 5,000 enemies with the Bow.

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Increase Damage dealt by 50% to enemies in close Range.
Range: 6m
*How to obtain: Kill 5,000 enemies with the Crystal Sword.

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Has 10% chance to remove an enemy's positive Defense when it takes damage.
*How to obtain: Kill 10,000 enemies with the Kunai.

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Your Damage scales with your Defense and Damage Mitigation.
Reduce Power damage scaling by 50%.
*How to obtain: Reach at least 1000 Defense.

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Increase Power damage scaling by 150%.
Multiply the Additional Projectile multiplier by 0.
*How to obtain: Kill 50,000 enemies with the Magic Wisp.

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Increase Defense Shredding by 9% for every enemy pierced.
*How to obtain: Kill 5,000 enemies with the Shuriken.

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Increase Defense Piercing damage scaling by 150%.
*How to obtain: Kill 50,000 enemies with the Worm Gland.

Talents that only effect projectile based weapons.

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Gain +1 Projectile every 20 levels, up to +8 Projectiles
*How to obtain: N/A

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Increase Projectile Damage by 10% for every enemy pierced, up to 1,000%.
*How to obtain: Kill 50,000 enemies with the Crossbow.

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Increase Projectile Lifetime by 45%.
*How to obtain: Kill 5,000 enemies with the Magic Wand.

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Weapons with the Magic tag get their Damage increased by 35% and Projectile Count increased by 2.
*How to obtain: Kill 20,000 enemies with the Shaman Staff.

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Critical Projectiles have infinite Piercing and increases their Defense Shredding by 30%.
(Additive).
*How to obtain: Kill 5,000 enemies with the Spear.

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Projectiles have 35% chance to have their Lifetime increased by 150% and have infinite Piercing.
*How to obtain: Kill 10,000 enemies with the Wind's Blade.

Talents that effect exclusively the area stat or melee weapons.

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Weapons with the Area tag gets their Attack Speed increased by 35%.
*How to obtain: Kill 20,000 enemies with the Arcane Beam.

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Increases Damage and Projectile Size of weapons with the Melee tag by 35%.
*How to obtain: Kill 5,000 enemies with the Cleaver.

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Increases Area Size by 45%.
*How to obtain: Kill 20,000 enemies with the Death Aura.

These talents only effect the player on kill and/or Overkill.

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Increases your Base Defense Piercing by 1 for every 500 monsters killed.
Killing an elite count as 100 monsters killed.
*How to obtain: N/A].

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On Overkill:
Permanently increases your base Experience Multiplier by 0.03%.
*How to obtain: Kill 50,000 enemies with the Katana.

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Overkilling an enemy applies 50% of the Damage dealt to the enemies behind in a Range of of 50m, at an Angle of 12°.
*How to obtain: Kill 10,000 enemies with the Pike.

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On Projectile kill:
The Projectile has 20% chance to spawn a Sub-Projectile if it can be split.
*How to obtain: Kill 20,000 enemies with the Swordrang.

Talents that effect the anachronistic ability which you unlock after killing a D-Rank Necromancer. (You will find it in the soul shop.)

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Increases Anachronistic duration by 275%
*How to obtain: Unlock Anachronistic.

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Increase Health Regen by 800% during Anachronistic ability
*How to obtain: Unlock Anachronistic.

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Increase Projectile Speed by 750% during Anachronistic ability
*How to obtain: Unlock Anachronistic.

Talents that give you cards or effect the cards that you receive.

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Every 15 level, your card selection only has [Heroic] Cards
*How to obtain: N/A

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Increase [Tainted] and [Ascended] card drop chance by 300%
*How to obtain: N/A

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Decrease [Weapons] drop chance multiplier by 50%.
Increase [Weapon Modifier] drop chance multiplier by 150%.
*How to obtain: Reach 40:01 in C Survival mode.

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Start the game with Fire Spirit at level 3
*How to obtain: Kill 100,000 enemies with Fire Ring.

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Start the game with Ice Spirit at level 3.
*How to obtain: Kill 100,000 enemies with Ice Nova.

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Start the game with Thunder Spirit at level 5.
*How to obtain: Kill 10,000 enemies with Thunder Staff.

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Start the game with Dash at level 3.
*How to obtain: Kill 10,000 enemies with Throwing Knife.

Talents that do not fit in with other groups.

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Every second, decays 1.10% of the Max Health of all enemies in a Range of 8m around you.
When the decay is active, you also lose 1.10% of your Heath per second as negative health regen.
If your Health is below 40%, the perk disables itself until you reach 70% of your Health.
*How to obtain: Kill 20,000 enemies with Blood Syphoon.

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Increases Artificer Potion's spawn chance by 250%.
Increases Artificer Potion's duration from 30 seconds to 60 seconds.
*How to obtain: Kill 50,000 enemies with Explosive Vial.

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Upon taking damage, you become immune to damage for 0.5 seconds.
*How to obtain: Kill 20,000 enemies with Thorn.

Talents that are exclusive to the summoner avatar.

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Every time a Summon dies, heal all other Summons for 10% of their Max Health.
*How to obtain: Kill 500 Living Mushroom with Summoner.

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Summons have 40% chance to shoot double projectiles.
*How to obtain: Kill 250 Goblin Mage with Summoner.

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Melee Summons gain 8% Damage every time they deal damage to an enemy.
*How to obtain: N/A

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Summons gain 8% Defense every time they take damage.
*How to obtain: N/A

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Increase Lifetime of all Summons by 30%.
*How to obtain: Kill 10,000 Golem Worker with Summoner.

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Upon dying, a summon deals 13% of an enemy's health as damage (ignores Defense).
Only deals 3% to Elites.
Does not affect Bosses.
*How to obtain: Kill 500 Living Cactus with Summoner.

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Reduce Damage Dealt to your Summons from Elites by 50%.
*How to obtain: Kill 4,000 Mummy with Summoner.

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Start the game with Fire Spirit at level 3.
*How to obtain: Kill 4,000 Kobold with Summoner.

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Increase Speed of all Summons by 30%.
*How to obtain: N/A

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Summons' projectiles have 40% chance to ignore enemies' Defense.
*How to obtain: Kill 3,000 Flying Scout with Summoner.

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Increase the Knockback of all Summons by 250%.
*How to obtain: Kill 1,000 Red Vulture with Summoner.

"Where did my stones go?"
Stones, as of the the 0.10 update, have been converted to Equipment, so they will not be included in this guide anymore.

If any of the information is either outdated, misspelled, should be explained better, straight up wrong, etc. Please let me know and I will fix it as soon as possible.