I just recently installed Debian 9.2.1 with an SSH server (No GUI)
But when I log in SSH via Putty, I get this weird message

"login as: nodrinks

****USAGE WARNING****

This is a private computer system. This computer system, including all
related equipment, networks, and network devices (specifically including
Internet access) are provided only for authorized use. This computer system
may be monitored for all lawful purposes, including to ensure that its use
is authorized, for management of the system, to facilitate protection against
unauthorized access, and to verify security procedures, survivability, and
operational security. Monitoring includes active attacks by authorized entities
to test or verify the security of this system. During monitoring, information
may be examined, recorded, copied and used for authorized purposes. All
information, including personal information, placed or sent over this system
may be monitored.

Use of this computer system, authorized or unauthorized, constitutes consent
to monitoring of this system. Unauthorized use may subject you to criminal
prosecution. Evidence of unauthorized use collected during monitoring may be
used for administrative, criminal, or other adverse action. Use of this system
constitutes consent to monitoring for these purposes.
Password:"

But when I log in with my password, it just prints out
"Access denied"

This happens until I get locked out for "Too many authentication failures"
It will work when I SSH locally, using "$ssh debian"

Possibly the keyboard layout on the server is different to your client? have you tried typing the password into something like notepad then pasting it into PuTTY?
edit: When my password is not accepted one of the first things I do is type it in the username field to ensure I am typing what I think I am typing.

Apparently all I had to do is use a bridged network adapter (I'm on VirtualBox)

What is "on VirtualBox"?
Please give some details of what is installed an how. Or, do you mean that this is solved by using a bridged adapter and whant to know what that means?

At face value what you posted does not make sense.

That weird message is a banner which is normally turned off by default. I can only assume that the hostname you were using was resolving to another IP address and obviously you could not login. A VirtualBox NAT adapter (default) acts like a typical home router. Without port forwarding you can not access a virtual machine. If trying to connect I would of expected a a no route to host or connection refused message.