If you received the generic Hotmail "Sign-in Error" message on Login screen, it may include a link whose target troubleshooting explanations by Microsoft and the Windows Live team, we archived on this page alongside clarifications. We'll put in layman's terms each of the statements discussed in Microsoft's official documentation for Windows Live Hotmail troubleshooting.
The Windows Live documentation starts by enumerating the four sign in error messages you may get, all of which are typically related to a technical issue on the mail servers (i.e., on Hotmail's side, not yours).
The Windows Live team's document goes on with "The server that hosts your account is currently undergoing a planned maintenance or experiencing a temporary problem. Please view the yellow outage banner on the top to see if there are any ongoing problems with the service. To determine which server you are on, view the "How to determine which server you are on?" article". This covers the most common reason for receiving any of the error messages when you try to login to your Hotmail account, namely a technical glitch, usually quite short, on Microsoft's end (remember, Microsoft owns Hotmail).
Finally, concludes the Windows Live help article on these generic Hotmail sign in errors, "Since this is only a temporary situation, there isn’t any action required on your part. You can try to sign into your account in a short time. Even if you have been seeing this error for more than 1 hr., it doesn’t normally last for more than 4 hours. If you cannot sign into your account for longer than 4 hours, please post your issue along with your account name in the Hotmail Sign In Issues forum".
In our experience, most of these problems "fix themselves" within the hour, so our recommendation is simply to wait 10-15 minutes, and go back to the Hotmail login screen - in some cases, note that if you have setup your web browser to automatically sign in to Hotmail (i.e., bypass the login screen), you will not even get a sign in error message. If this is the case, make sure that you try signing in from a browser you have used before to check your email messages.
Unlike the recommendations given by the Windows Live team at the end of that tech support page, we do not recommend that you change your browser settings: if you have been able to login to your Hotmail account in the past from this computer and web browser, your settings themselves are fine (assuming no-one changed some options unbeknownst to you - see "Sign in to multiple / different Hotmail accounts on the same computer" on how to share the same PC / Mac for multiple Hotmail users).