Specializations are an endgame feature in The Division 2. The Specializations add new bonus weapons to your inventory. They also give you other passive buffs and active skills.

How to Unlock Specializations in The Division 2:

Reach Level 30 and finish the Story (conquer the Capitol Stronghold)
Talk to the Quartermaster in the White House. A new desk becomes available at the Quartermaster where you can equip the Specializations.
Pick between the Demolitionist (Grenade Launcher), Survivalist (Crossbow), Sharpshooter (Sniper)

How to Use Specializations in The Division 2: Hold Triangle (PS4) / Y (Xbox One) to equip the Specialization weapon. Some of the specialization skills are passive. Only their respective weapon needs to be equipped.

There are 3 types of Specializations: Demolitionist (Grenade Launcher), Survivalist (Crossbow), Sharpshooter (Sniper). You can switch between them at any time at the Quartermaster. They are not permanent. So don't worry, you can test it out and if you don't like it change to a different class.

How to Get Ammo for Special Weapons:

Grenade Launcher 40mm Grenade = Kill enemies with explosives (e.g. grenades)
Crossbow Explosive Bolts = Kill enemies with weapons while they are under a status effect (e.g. while they are on fire or while they are electrified)
TAC-50 C Sniper Rounds = Kill enemies with headshots

To upgrade your specials you must earn specialization points which you get from doing invaded endgame missions (marked red on map). It's a long grind to fully upgrade everything.