When you click the Register link you're presented with this webform:

If you're in a hurry and click Register your Application you'll find this error:

The fields Name, Description, Website URL and OAuth Domain are required.

You can probably dream up a name and a description but those pesky Website URL and OAuth Domain keep bugging me. Here is the shortest route to success:

And now click on the big blue Register your Application.

Congrats, you now have registered an app. You find the key as the third item on your app registration page:

If you scroll all the way down you'll find a blue Edit this App button, allowing you to change any of the values you previously entered. So if you later do get an website or read-up on Oauth flow you can adapt your values accordingly.

Enjoy your new app and key!

^{credits for bad.invalid goes to Kevin Montrose in his answer}

When you click the Register link you're presented with this webform:

If you're in a hurry and click Register your Application you'll find this error:

The fields Name, Description, Website URL and OAuth Domain are required.

You can probably dream up a name and a description but those pesky Website URL and OAuth Domain keep bugging me. Here is the shortest route to success:

And now click on the big blue Register your Application.

Congrats, you now have registered an app. You find the key as the third item on your app registration page:

If you scroll all the way down you'll find a blue Edit this App button, allowing you to change any of the values you previously entered. So if you later do get an website or read-up on Oauth flow you can adapt your values accordingly.

Enjoy your new app and key!

^{credits for bad.invalid goes to Brock Adams in his answer}