Natural weapons are just weapons that are a part of the body and have no special rules
Natural weapons are most commonly found on monsters and the best description for them in general is found in the intro to the Monster Manual (p. 10-11) or the corresponding portion of the basic rules:
The most common actions that a monster will take in combat are melee and ranged attacks. These can be spell attacks or weapon attack, where the "weapon" might be a manufactured item or a natural weapon such as a claw or tail spike.
So it is simple, natural weapons are, as their name implies, weapons that happen to be naturally occurring on a monster or creature's body.
Most of the time in monster stat blocks especially they aren't even labeled as natural weapons in any special way. For example, the crab has:
Claw. Melee Weapon Attack: +0 to hit, reach 5 ft., one target. Hit: 1 bludgeoning damage.
Obviously the crab's claws are not manufactured and are thus natural weapons. And the rules simply call it a "melee weapon attack" since there's no difference in rules that would make it necessary to point it out.
When you see a natural weapon, you simply treat it exactly the same way as any other weapon for the purposes of attacking. Nowhere in the rules are there any default rules to treat them differently in any capacity. The only implied difference between natural weapons and manufactured weapons is that natural weapons cannot be easily removed (so disarming a monster would be difficult if not impossible). This is an area of the rules not explicitly laid out though so your DM will have to make specific rulings if any conflict comes up.
When a PC encounters natural weapons it is through a specific feature that gives them one (it could be a spell or magic item or racial feature or something else). That feature will describe how any special rules surrounding the natural weapon (if any) including how much damage it does.
Some natural weapon abilities allow them to be used to make unarmed attacks
Even though natural weapons are weapons, there are many examples of abilities granting the power to use natural weapons to make unarmed strikes (something that normally can only be done with no weapon).
For example, the tabaxi race has the Cat's Claws trait (VGtM, p. 115), which says:
[...] your claws are natural weapons, which you can use to make unarmed strikes. If you hit with them, you deal slashing damage equal to 1d4 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
Again, if an ability granting a natural attack has any special rules or powers it will say so (just like the above example). But this is not a general rule. In general, natural weapons are just normal weapons.
The insignia of claws simply enhances natural attacks
The description of the insignia of claws says:
While wearing the insignia you gain a +1 bonus to the attack rolls and the damage rolls you make with unarmed strikes and natural weapons.
It is good to note that the insignia does not give you natural weapons, just enhances them if you happen to have them.