Natural weapons are a kind of weapon...
There are almost no general rules on natural weapons in the PHB or Basic Rules describing what they are and how they interact with other rules. Some game features grant specific natural weapons to a character, and some monsters/NPCs have specific natural weapons listed in their statblock, but we can't derive general rules from that.
As of 2021, the Sage Advice Compendium (v2.7) contains a single official ruling about natural weapons:
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.
This is a pretty clear official ruling: Natural weapons are weapons. This is a logical interpretation of the rules, given that they're referred to as "weapons" in the first place.
On the topic of unarmed strikes:
As clarified in the Q&A Can natural weapons be used for unarmed strikes?, natural weapons are not used to make unarmed strikes unless a game feature explicitly states otherwise. By default, natural weapons are just another kind of weapon (separate from simple and martial weapons). However, as noted in my answer there, most features that grant natural weapons to player characters do specify that the natural weapon can be used for unarmed strikes.
...but they don't have any weapon properties unless they say they do
In an unofficial tweet from June 2015, rules designer Jeremy Crawford responded to a since-deleted tweet asking whether natural weapons have the finesse property:
Are natural weapons considered finesse weapons? A lot of them use dex for to-hit and damage, but I can't find a reference.
No. A weapon has the finesse property only if its description says so, and using Dex. to hit doesn't equal finesse.
In general, weapons don't have specified properties (e.g. finesse, light, thrown, etc.) unless their descriptions in the rules say they do. As such, natural weapons don't have any weapon properties unless they say they do.
Whether they're melee or ranged weapons is indicated by the kind of attack they can be used to make, as detailed in their description – but "melee" and "ranged" aren't weapon properties, just classifications of weapons. Beyond that, we can't assume they have any weapon properties unless explicitly stated otherwise.
Most features don't care whether you're using a weapon (or what properties it has) anyway
Natural weapons are usually used to make melee weapon attacks. (Occasionally, the description of a natural weapon specifies that it is used to make ranged weapon attacks, though I think that mostly applies to monsters that have natural weapons.)
In general, most game features that apply to or interact with an attack don't care whether you're using a weapon or not, and very few features rely on specific weapon properties. Many attack-related features trigger off "weapon attacks" or specifically "melee weapon attacks", rather than requiring a weapon or relying on the weapon's particular properties. (The rogue's Sneak Attack is one of the few notable exceptions, requiring a finesse or ranged weapon.)
As such, even if natural weapons weren't considered weapons (and regardless of whether they have any particular weapon properties), most such features would work with natural weapons anyway.