From Diablo Wiki
Common errors when trying to connect and play Diablo III on Battle.net.
"This Battle.net account does not have a Diablo III license attached to it."
This is an error associated with Guest Passes. Check that your game region is set to the same as your account region. To do this, launch Diablo III. Do not log in but instead go to 'Options'. Select 'Account'. From the dropdown box marked 'Server Region Selection' select the correct server.
If this doesn't work change the password on your Battle.net account and this will force your account to refresh.
The server is full. This is likely due to high login traffic. The only solution is to keep trying to log in.
"There is a temporary outage of the Battle.net Service. Please try again later."
This error will occur when the servers are under a heavy load. Try logging in again after 5-10 minutes. It may take several attempts before you are able to successfully connect.
While Customer Support is unable to assist with this error, it is a known issue that the development team is working to resolve.
This is due to an incomplete file having been downloaded. Check the download directory to see if there is a .prt file in there. If so, restart the downloader and allow it to finish.
Server is down usually just prior to patching.
"The request has timed out." This is an issue Blizzard are working to resolve. Customer Support cannot assist with this error. It is recommended that you powercycle your router and modem as detailed in Step 2 of the Diablo III Connection Troubleshooting page. (Blizzard)
Can occur in Diablo III when a computer is unable to maintain a stable connection to our Battle.net servers. The Battle.net connection runs in addition to other connections to the Diablo III game servers. Many players can work around the problem with the following steps. (Blizzard)
Auction House error that is throttles the item listings being created at the same time to improve stability. It's not a permanent error and you should trying listing your item again later.
"There was an error downloading agreements" One possible fix comes from Trudey
- This error is caused due to a problem with the cache folder that battle.net created on your computer. To fix this open the “C:\ProgramData\Blizzard Entertainment\Battle.net” folder on Vista or Win 7. Windows XP users can find this at “C:\Documents and Settings\All Users\Application Data\Blizzard Entertainment\Battle.net”.
- You should see a folder called “Cache”. Rename this folder to any other name or delete it. Now start Diablo 3 and the error should be fixed.
Specific to the PTR server. This occurs when a player tries to copy over characters from the live server to the PTR more than once in 72 hours. 
This error typically means that your username or password has been entered incorrectly. Check that you are correctly typing your username (your Battle.net email address) and password. Take care not to add a space after your email address.
Before following the steps below, change your registered email address back to the previous email address and try again if it was recently changed. Please also try resetting your password. To reset your password, use the Login Support page.
Unknown but it usually comes after a disconnect from the game due to the player's loss of internet connection. Sometimes when trying to immediately resume the game an Error 3007 can occur.