The answer on this depends on the audience for your Web API and what you want to authenticate exactly.
- Do you want to authenticate a client application that uses your Api?
- Do you want to authenticate a user from your application to retrieve their data within a client application (using your Api)?
- Or do you want to authenticate both the client application and the user using the client application.
Depending on what you want to authenticate you have multiple options. But always keep in mind that it is better to go with a solid solution where many client libraries are available than reinvent you own. Never do a little off this, but in your own way, chose one way of authentication, stick to it and don't break the client libraries.
Basic authentication: Is very easy to implement, but you authenticate a client app with it, not a user. This kind of authentication is nice when business trust relation needed and authentication and safety is not your very first concern. But there is no way to track a call in your API back to a certain user, just a client application. Of course you could save your user's username and password in a client application but this is a bad practice in more then a single way.
Token based authentication: Their are many ways of token authentication but the one i'm talking about here is a single token for a user which the user copies to the client application to get access to your Api. This way you can authenticate a user (who made this call in my Api?) And it is fairly easy to make and use. The withdrawal is that it is not the most secure way, requires user interaction and that a user possibly uses his Api token in more then one application. You could extend this way of authentication with basic authentication to authenticate a client. So a clientid + clientsecret + token to identify the user. But I think if you want to accomplish this it would be better to take a look at Oauth2.
OAuth2: If you want to have full access over your authentication you can go this way. It is possibly the most future proof way to go, but also requires the most work (at least at the identity provider/resource provider side. The client application has a fairly easy time implementing this with a lot available client libraries. If you go with this way of authentication (also token based) you can authenticate the client and the user, without the need to share your users username and password.
My recommendation: would be to go with Basic Authentication if this fits your case, it is easy and together with HTTPS is fairly safe. If it doesn't fit I would go with Oauth2 because it is the most solid and used standard (Instagram/Google/Facebook), gives you a lot off freedom and with a growing ecosystem gets easier and easier to implement. After all for someone implementing your API it is way more interesting to learn something about Oauth 2.0, then learn about the jgauffin way of doing things.
Reference: I would also like to invite you to have a look at Apigee's website. Api's are their business and they have quite some interesting reads. One of them is is a free ebook - Oauth the big picture which also has a interesting paragraph where they ask if you really need Oauth. (From page 16 - Is OAuth all you need for API security?)
For server-to-server APIs - APIs designed to be used only by a small number of servers –
OAuth is overkill. Having a separate set of authentication credentials for each app is a nice
feature of OAuth, but for server-to-server use, the need to log in securely using a browser,
or to implement other steps in the OAuth “dance,” gets in the way.
Instead, using a simple security standard like HTTP Basic authentication and assigning a
unique password to each app is sufficient. Two-way SSL is another good, albeit
cumbersome approach that has the advantage of stronger, more traceable authentication.
However, think ahead! Are those APIs really only going to be used by servers forever?
Exisisting Solutions: Whatever way you go leastprivilege - Dominick Baier and his nuget packages can give you a nice headstart. Implementing basic authentication using his Identitymodel is really easy. Also if you want a ready-to-go identityserver to provide you with tokens look at his identity server which does all you can think off. However if you decide to go for Oauth2 I would also have a look at DotnetOpenAuth since it is (imho) a bit more configurable and easier to tweak to your own like, but it also takes more work.
