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What is the relationship between OAuth2 and MFA

This is my first question on the Stackoverflow. I have already read the https://stackoverflow.com/help/how-to-ask section and I have read and read and read over the internet to find the connection between OAuth2 and MFA. Which made me more confused. When I take them separately I understand the basic concept.
Background. I am developing a windows application and using mailkit to implement the email there. When I research how to implement MFA on that, most of the answers are talking about OAuth2. That's why I really need to find an answer for this questions.
I have read
https://learn.microsoft.com/en-us/answers/questions/513048/microsoft-office-365-outlook-sending-emails-using.html and https://github.com/jstedfast/MailKit/blob/master/ExchangeOAuth2.md and many more articles to achieve my goal. They all are talking about OAuth2 not MFA.
I am sorry if my questions doesn't make much sense. Please help me to understand this.
Thank you.

OAuth 2.0 and MFA are not related to one another directly.
OAuth 2.0 is an authorization protocol which provides a client application with the delegated access. Way of delegation is defined by grant type and based on the use case you can choose one or another. It does not tell you directly how to authenticate the end user.
For example, using classic authorization code flow you can point the user to login page and optionally consent screen (allow / deny access to the app). An example of URL would be:
https://authorization-server.com/auth?response_type=code&client_id=CLIENT_ID&redirect_uri=REDIRECT_URI&scope=photos&state=1234zyx
Then it's up to authorization server how to authenticate the end user. It can be a login / password screen followed by any type of MFA.
MFA, on the other hand, stands for multi-factor authentication, meaning that there can be 2 of more factors of authentication (such as login and password pair followed by one time code, SMS or anything else).
Based on my understanding of the OAuth2 with Exchange that you've shared, you just need to register an application on Microsoft platform in order to access Microsoft APIs for email sending. That's indeed an OAuth 2.0, but I don't see anything related to MFA there.

Authenticate domain user after connection to domain is inaccessible C#

We have an MVC application that validates windows users by instantiating a principalcontext, locally and for domain users. However, this immediately fails when a connection to the domain cannot be made. Is there a way to leverage the capability of windows to still validate domain credentials when disconnected from the domain?
We see that in SSMS you can also use windows authentication after the server has lost access to the domain.
To be clear, the machine is joined to the domain but does not have access to the domain controller, i.e. a corporate laptop that is taken home.
Thanks for the help in advance.

The answer is in how Kerberos (and I believe NTLM is similar) works in that they use session tickets. So once authentication is successful against a domain controller, you have a "ticket" that proves you authenticated. When you authenticate to anything else that requires Windows authentication, the ticket is sent.
To take advantage of this, you must use the built-in Windows authentication. You cannot take a username and password and try to authenticate them that way.
If everyone who uses your website can use Windows authentication - then you can enable it for your whole site.
If you have a mixed audience - some who have a domain account and some that don't - it's a little trickier, but still doable. I've done it. You can look at the OWIN-MixedAuth project. I haven't used that specifically.
That project seems to use a separate button for the Windows authentication. When I did it, I made it seamless (it tries Windows auth and fails back to a login page). I did that by doing an AJAX request in the background to a page that requires Windows authentication, and if it succeeds, just forward on. If it fails, show the login fields.
All that said, I don't know it will work if the server cannot reach the domain. It will probably work for users who have already authenticated to the site before it lost connection to the domain, or if the site is hosted on the same computer that it is being accessed from. But it might not work if a user it has never seen before tries to login while there is no access to the domain. You will have to test.
But the benefit of using Windows authentication anyway is that you can make the login seamless. As long as the site is in your Trusted Sites (in the Windows Internet Options) then IE and Chrome will automatically send the user's credentials.

How can I authenticate a windows user over a REST API call without IIS/WCF?

While developing an on-premise, intranet-only REST API server for my company, I managed to completely confuse myself regarding authentication issues.
I have:
A REST web server implemented in C#, using self-hosted Nancy, running as a Windows Service
A C# client that can query the server, run by a user in our company
I do not have:
Any form of access to our Active Directory and/or domain controller, apart from what any application running under Windows normally has
Any way to influence AD settings or configuration
Active Directory Federation Services (ADFS) (I think. We use Windows 7 and Office 2010, just to give some perspective on the state of the software landscape)
Azure Active Directory (AAD)
I want:
A way for the server to authenticate that a request is made by a user of our company
It is perfectly fine if the client has to sent some additional authentication data with each request, as long as it does not contain the user's password in any form
I do not want to:
Have to set up any additional software (my server must be minimum configuration and maintenance, so the average user can install and run it)
Install / configure / maintain an IIS server (see above)
Use ASP.net (way too big for my needs, plus see above point(s))
Handle user passwords in any way (company policy and common sense)
Impersonate the user (I only need to validate the authenticity of the request)
Implement my own user account database. We already have half a dozen services that need their own username/password combinations, I do not want to add yet another one
I have read articles that show how to use Windows authentication with IIS, or how to use Azure Active Directory (AAD) with Nancy. Other questions here have already informed me how to authenticate username / password combinations against the Active Directory. However, none of these satisfy all of my requirements or have requirements of their own (like AAD/ADFS) that I cannot meet.
It seems that Kerberos/SSPI might be what I want, but it seems very involved and quite complicated to get working with C#. It is possible I will have to go this route, but I could really benefit from some minimal working example (the accepted answer provides a C# implementation/wrapper, including an example project, but I can't seem to be able to make heads or tails of it).
Maybe I am naive, but what I image the solution to be is something along the following lines:
The client queries a service (AD, Domain controller, ...?) for some form of authentication token, using the credentials of the currently logged in user
The token is then sent to the server together with the username, as part of the request that needs to be authenticated
The server extracts the token, and queries the same service (AD, Domain controller, ...) whether the token is authentic, valid and belongs to the user in question
Is this possible at all? Ideally, with some sort of ready made library that I can plug in to my projects (I'm reaching, I know)?

You can do this with stateless authentication and Jwt. Send a username and password to "/ auth" (example) and "/ auth" will search the AD (example) and validate if the user exists, then create a Jwt token with the name of the user on load. When you make a request, you will only send a Jwt token and Nancy validates the token.

Authentication using Azure

I have an Azure account and currently a Mobile Service setup with a SQL Database so that my Windows Store app can communicate with the database.
I have developed sites using ASP.NET WebPages authentication. And I need something similar for my Windows Store app.
I have successfully gone through the documentation and tutorials on the Windows Azure website and implemented ACS (Windows Live ID, Google, Yahoo!, and Facebook) - but the thing is - I don't want Google, Yahoo!, Facebook or even Windows Live ID or Microsoft Account) logins - I want my OWN login but it seems that they don't give you this option (correct me if I am wrong).
I need to allow users to signup from within my application (that means, providing their name, DOB, email, phone, address, etc) and shove it all in my database.
Now, after implementing Microsoft Account login with my Azure service, I found out that you can't even get the most basic information about any user who has logged in to your application - not even an email address.
I have spent hours searching online for something that could possibly help but I am running out of keywords - and have not hit a single related result yet.
Does anyone know if this is possible? How would we go about integrating login and signup with a Windows Store app that set/gets this data into/from a Windows Azure service?
Any code, samples, links, tutorials, documentation, etc would be highly appreciated.

You have gone down the road of hooking up external identity authentication, which in my opinion for an external facing web application is a better approach. Benefits are:
Your application is only responsible for Authorization not Authentication. There is a whole lot of work involved in Authentication and a large number of best practices. Best let those who know best take the burden of this. This doesn't mean you shouldn't try and understand it though.
If your site gets hacked you don't have to tell them that their username / email and password combo has been compromised and they will probably have to change there passwords on other sits.
You are also making sure that your users don't have to remember / manage yet another username / email address password combo
If you really want to do the Authentication then that is fine but you will need to do it yourself. Have a look at examples on Asp.Net Membership. This is not the only way and nor is it the best way but there are lots of examples.
Now if you decide you want to use external authentication I can give you some pointers to help with your current implementation.
First thing to note that the Id you get back from Live, Google, Facebook can only be assumed to be unique for that provider. Therefore if you want to keep a profile in your system for that identity and you want to use more than one provider you will need to implement it in such a way that you can keep the id unique in your system and help you associate it with a provider.
Website Authentication with Social Identity Providers and ACS Part 2 – Integrating ACS with the Universal Profile Provider
As you have found out not all of the Authentication providers return the same "claims". A claim is something that user claims to have, such as an email address, name, date of birth, etc. All the ones you can use by default via the ACS return Uid and some return a name and email address. What you have to do is fill in the gaps. When someone registers you will need to pull the relevant claims and then ask them to fill in the missing ones. You may also want to map the different claims in the ACS to a common name that you can use in your app as one provider might use slightly different names.
Federated Identity with Windows Azure Access Control Service
Just because you do not handle Authentication you still need to be responsible for keeping your application secure. Half of the work has been done for you so your code should be a lot lighter but you will still need to make use of roles.
Windows Azure Role Based authentication (ACS)
The really nice thing about this approach is you can implement your application the same why SO have done with there identity model. You can allow users to associate multiple identities against their profile meaning they can login how they want to.

If you choose not to use the built in providers for ACS you will need to implement your own Identity Provider using SAML, OpenId, etc...
You can look into the Windows Identity Foundation (WIF) for implementing WS-Trust or WS-Federation.
There is also ADFS which has the same set of support but uses Active Directory with WIF and Azure has its own version of AD that can be used.
There is also thinktecture identityserver which can jumpstart your venture into IdP land, but I have not used it myself yet.
If you want to go the OpenId route there is DotNetOpenAuth.

If you're looking to add custom identity to your Mobile Services app, check out Josh's post on custom auth: http://www.thejoyofcode.com/Exploring_custom_identity_in_Mobile_Services_Day_12_.aspx

Windows Authentication SSO in ASP.NET Remote Web Application

There is a requirement to implement Single Sign On (SSO) in our website so that users accessing it from the intranet won't have to type in their credentials.
The problem here is that the website is going to be hosted on the internet, on a remote server.
Is there some way this can be done?
--EDIT--
I looked at following link:
http://en.wikipedia.org/wiki/Active_Directory_Federation_Services
And the example scenario that is mentioned in there is exactly what we require.
Here is the detailed explanation on the situation:
There is an Active Directory Domain Controller that is used to authenticate users in the intranet.
Once the user logs into the machine, and opens up the remote website, the website should somehow verify that the user is already logged into the intranet using AD credentials and automatically allow access to the website.
Also, the website is supposed to be getting a Security Token that can be used to authenticate the user.

Of course, for example, Microsoft's way of doing that for connecting Office365 to your Intranet is called Active Directory Federation Services.
It is (as most single sign on solutions) not entirely straight forward, and it assumes a domain on your intranet, but since Office365 uses it, it is and will most likely continue to be well supported in the future.

It's just a matter of integrating your web app with the authentication provider that you are using for SSO. The details are going to vary dramatically if your using OAuth vs. Shibboleth vs. ADFS vs. etc, so there really isn't enough information in your question to give a helpful answer.