I am seeing some magical items that refer to natural weapons, but I can't find information about what they actually are. I don't remember seeing it in the Player's Handbook, and googling it isn't finding anything. Can anybody tell me what they are, and where the rules for them are found?
If it means anything, the item I was seeing this on was the Insignia of Claws from the Hoard of the Dragon Queen campaign source book (described on p. 94).
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.
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 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.
This is covered in the 2020 edition of the Sage Advice Compendium:
Are natural weapons considered weapons? Things designated as weapons by the rules, including natural weapons, are indeed weapons. In contrast, unarmed strikes are not weapons. They are something you do with an unarmed part of your body.
Consider the Lizardfolk, whose Bite feature is expressly described as a 'Natural Weapon':
Your fanged maw is a natural weapon, which you can use to make unarmed strikes. If you hit with it, you deal piercing damage equal to 1d6 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
—Lizardfolk, Volo's Guide to Monsters, pg. 111
Note that last part of the description: normally, an unarmed strike simply deals 1 + STR damage, and it's incumbent on special Class features, Racial features, or Feats to be improved. A Natural Weapon is one such Racial feature, like with the Lizardfolk Race. In essence, Lizardfolk are allowed to use their Bite as an unarmed strike, dealing 1d6 + STR instead of 1 + STR; but this special feature also constitutes a Natural Weapon.
Some other races have similar features:
Your horns are natural melee weapons, which you can use to make unarmed strikes. If you hit with them, you deal piercing damage equal to 1d6 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
—Minotaur: Horns, Guildmaster's Guide to Ravnica, pg. 18
Your hooves are natural melee weapons, which you can use to make unarmed strikes. If you hit with them, you deal bludgeoning damage equal to 1d4 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike.
—Centaur: Hooves, Guildmaster's Guide to Ravnica, pg. 15
Because of your claws, you have a climbing speed of 20 feet. In addition, 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.
—Tabaxi: Cat's Claws, Volo's Guide to Monsters, pg. 113
So in the case of the magic item you cited, Insignia of Claws, a Lizardfolk using that insignia would be able to make this attack with a +1 bonus to hit and damage.
This is relevant because Druids gain the ability to shapeshift into various beasts while still retaining a wide variety of class features. As a consequence, Unarmed Strikes made while Wild-Shaped may-or-may-not qualify to be used in conjunction with the attacks provided by their wild-shaped form.
From the Basic Rules,
Melee and Ranged Attacks The most common actions that a monster will take in combat are melee and ranged attacks. These can be spell attacks or weapon attacks, where the “weapon” might be a manufactured item or a natural weapon, such as a claw or tail spike. For more information on different kinds of attacks, see chapter 9 of this document.
Anything that is normally attached to the body of a creature is a natural weapon. For example, take a Satyr. It has a ram attack, a shortsword attack and a shortbow attack. That ram is a natural weapon, the other two are manufactured.
Note that "weapon" has a bit of a fuzzy definition since creatures have these "weapons" listed as actions rather than equipment.
