Authentication and Authorization in asp.net core WEB API

Basic auth is as the name suggests, very basic and not very secure, it uses base64 encoding of the username and password so you must use HTTPS if you use it, but best is not to use it at all.

A bearer token is a type of token which effectively gives access to a resource to the "bearer" of the token. Both basic and bearer are used in an HTTP Authorization header.

You can have different formats of bearer tokens, one of which is JWT - JWT is the industry standard so I recommend you use it, and therefore you'll be using bearer tokens.

This article is a good starting point to look into all this in the context of asp.net core. See also this video series and this article goes into more detail about JWT validation.

Edit

To answer your questions in the comments:

OAuth is a standard for users to delegate permissions to apps or websites to access their resources, for example when you allow some web app to post on your behalf to your Facebook feed. Various tokens are used in this process and they're very often JWT. OAuth2 adds authentication via OpenID Connect.

OWIN on the other hand is a standard for web servers which decouples IIS and ASP.NET with the aim of allowing ASP.NET to run on other web servers which implement OWIN and other frameworks generally to run on OWIN compatible servers if those frameworks are OWIN compatible.

Auth0 is an identity platform which can do OAuth and allows you to use JWTs, generally it handles your identity and SSO. IdentityServer is another identity platform with some similar features.

I'd still recommend starting with the articles I linked at the top, don't worry too much about OWIN, and read more about OAuth to determine if you really need it. If you do, I'd recommend IdentityServer.