In Extinction mode, there is a "currency" called Teeth. You can earn teeth through various tasks: completing Extinction missions, using relics, playing in hardcore mode, helping other players complete a mission for the first time, and every 300 cryptid kills (also there is a "bonus pool" that refills every 24 hours giving you 1"freebie" teeth for simply progressing past the first area of hives on any map).
EDIT - New Ways to Earn Teeth:
+ 4 teeth per game for completionist
+ 2 teeth per game for completing 10 challenges (i.e. not failing more than 2 challenges)
+ unconfirmed: platinum, no downs, no drill repairs, ...
In your loadout in Extinction mode, there is a section at the top that says "Armory: Get Upgrades" - this is where you spend your hard earned teeth for class and equipment upgrades.
They implemented the Armory for DLC 2, but have since updated it with DLC 3, and DLC 4. Here is the master list:
(key: 1/5 = wait until you've got everything that's better, 3/5 = it's useful, 5/5 = it's necessary, and cyan if I haven't tried it & I'm reporting from someone else's experience)
The master list is in the order of the list in game.
Name: Faster Revive
Cost: 10 teeth
Upgrade: Shortens the time it takes for you to get revived.
GRILL rating: 3/5
Name: Start with a Locker Key
Cost: 40 teeth
Upgrade: Start with a Locker key on missions with Lockers.
GRILL rating: 5/5
Name: More Cash
Cost: 20 teeth
Upgrade: Start with an extra $1000
GRILL rating: 1/5
Name: Flaming stowed Riot Shield
Cost: 20 teeth
Upgrade: A stowed Riot Shield will burn enemies while blocking attacks from the rear.
GRILL rating: 5/5
Name: Higher Quality Incendiary Ammo
Cost: 5 teeth
Upgrade: Incendiary Ammo does more total damage.
GRILL rating: 3/5
Name: Electric Knife
Cost: 30 teeth
Upgrade: Electrically charges the knife so that a melee will arc electricity to nearby Cryptids, stunning them. The Electric Knife has a recharge time of 4 seconds.
GRILL rating: 3/5 (If you have some extra teeth lying around, this is a fun upgrade. The only down side is regardless of whether you get the electricity or not, you have to wait 4 seconds after every melee. It also seems to work sporadically in tandem with body armor, i.e. when attacked you emit an electric blast that stuns the aliens.)
Name: Sniper Deadeye
Cost: 40 teeth
Upgrade: Sniper Rifle hits are always critical and kills will earn more cash.
GRILL rating: 5/5 (fantastic, earns more money and cryptids go down super quick)
Name: Seeker Explosion Protection
Cost: 5 teeth
Upgrade: Take less damage from exploding Seekers and Bombers.
GRILL rating: 1/5
Name: Toughness
Cost: 25 teeth
Upgrade: Flinch less when taking damage.
GRILL rating: 5/5 (This upgrade is a real bargain for 25 teeth, and it greatly reduces the flinching when dealing with those pesky pods on Mayday. It's great for the other maps as well - think about it, what do you spend the most time doing in a game? Looking down the site. Get this upgrade. )
Name: Hypno-Knife a Rhino
Cost: 50 teeth
Upgrade: Players who use a hypno-knife on a Rhino can turn him into a valuable ally.
GRILL rating: 3/5
Name: Start with a Skill Point
Cost: 20 teeth
Upgrade: Start each game with an extra Skill Point.
GRILL rating: 5/5
Name: Faster Health Regeneration
Cost: 5 teeth
Upgrade: Regenerate health faster.
GRILL rating: 1/5
Name: High Grade Armor Piercing Ammo
Cost: 5 teeth
Upgrade: Armor piercing ammo will inflict increased damage when impacting a Cryptid's armored head plate.
GRILL rating: 1/5
Name: Damage Upgrade
Cost: 10 teeth
Upgrade: Increases the damage of all attacks by 5%
GRILL rating: 0/5 (don't even bother, not noticeable)
Name: Stun Armor
Cost: 10 teeth
Upgrade: With armor equipped, this will periodically stun attackers.
GRILL rating: 1/5
Name: Sentry Armor
Cost: 5 teeth
Upgrade: Sentry guns you place can sustain more damage.
GRILL rating: 1/5
Name: Medic Upgrade
Cost: 15 teeth
Upgrade: The Medic is surrounded by a therapeutic energy field. Allies inside the field are healed or revived from last stand.
GRILL rating: 5/5
Name: Engineer Upgrade
Cost: 15 teeth
Upgrade: The Engineer lays down a Stasis Field that both protects from Scorpion and Seeder attacks and creates a safe zone that Cryptids will not enter.
GRILL rating: 3/5
Name: Weapon Specialist Upgrade
Cost: 15 teeth
Upgrade: The Specialist has an adrenaline rush, greatly increasing damage done and keeping his clips completely topped off. Nearby allies also receive a modest damage boost.
GRILL rating: 5/5
Name: Tank Upgrade
Cost: 15 teeth
Upgrade: All Cryptids begin attacking the Tank, who is now invulnerable and able to stun enemies with melee attacks.
GRILL rating: 3/5
Name: I.M.S. Fast Trigger
Cost: 5 teeth
Upgrade: Shortens the time it takes the I.M.S. to detect nearby enemies.
GRILL rating: 3/5
Name: Keep Valid Attachments
Cost: 10 teeth
Upgrade: Attachments transfer if they are valid for the new weapon.
GRILL rating: 5/5
Name: Magnum ACOG attachment
Cost: 2 teeth
Upgrade: Magnum pistols will have an ACOG equipped.
GRILL rating: 1/5
Name: I.M.S. Flaming Pools
Cost: 10 teeth
Upgrade: When equipped, the I.M.S. explosions will leave lingering fire pools.
GRILL rating: 0/5 (hurts you and your teammates, don't waste your teeth)
Name: No Snare
Cost: 10 teeth
Upgrade: With this upgrade, Hunter and Seeker Cryptids will no longer snare players with their attacks.
GRILL rating: 3/5
Name: Pistol Ark Attachment
Cost: 125 teeth
Upgrade: Replaces the Muzzle Brake on your upgraded pistol with the Ark attachment (basically PaP's it, shoots lasers).
GRILL rating: 5/5 (This is basically like starting out with Mustang & Sally's every zombie match. The pistol ark's are fantastic, and make the pistols only relic a piece of cake for any map.)
Name: Class Upgrade Frequency Boost
Cost: 200 teeth
Upgrade: Lessens the cooldown time between uses of the Class upgrades.
GRILL rating: 3/5 (Very great if you have some teeth lying around, 200 of them at that. A bit overpriced if you ask me, but it does it's just damn well.)
Name: Vulture Extended Duration
Cost: 50 teeth
Upgrade: Extends the duration of the Vulture and increases the frequency of the +4 Vulture firing incendiary rockets.
GRILL rating: 3/5 (the vulture is a powerhouse on it's own, naturally making it stronger makes it more badass - but it's not necessary)
Name: Master Scavenger
Cost: 200 teeth
Upgrade: Scavenge unique items such as Extended Mags and the Ark attachment as well as more powerful Hypno Knives and Flares.
GRILL rating: 3/5 (If you have extra teeth then go for it - but there appears to be a very inconsistent nature involved with getting ark. I'm not going to spend my hard earned teeth just to get ark's sometimes... I want at least one ark every game.)
Name: Cryptid Slayer Ammo
Cost: 500 teeth
Upgrade: This special ammo has the combined firepower of all of the existing ammo types.
GRILL rating: 3/5 (seems pretty cool, but I slay fine as is )
Name: Arcing Stun Ammo
Cost: 50 teeth
Upgrade: Stun ammo will arc electricity from one Cryptid to another in close proximity.
GRILL rating: 1/5 (stun ammo is lame, end of story. some people swear by it - I don't)
Name: Double Class
Cost: 1000 teeth
Upgrade: Choose any two Classes with this upgrade. Note: Skill points will be applied to both classes when applied in game.
GRILL rating: 3/5 (the only reason this isn't "5/5" is because of the cost. No "average" player could attain this upgrade, and it's silly for IW to include such an unrealistic goal. Only those who have grinded 1000's of escapes will be able to achieve "Double Class", after those thousands of escapes - how much more are you really going to want to play?)