OS X 10.8.5 is Mountain Lion. In the future posting in the correct forum gets the best results.
Under Windows in the Menu bar select Connection doctor. You will see green to indicate a connection and red to show there is a problem. You can click on more details button, but with multiple accounts this is often hard for some users to read result. Except for iCloud account you'll see two entries for each account 1) receiving and 2 SMTP for sending.
Sending issues usually involve SMTP settings.
EDIT SMTP SETTINGS
In Accounts window > Outgoing Mail Server SMTP: select to Edit SMTP Server list... from popup
Select account in the list.(It's not uncommon to find extra SMTP servers listed. When you remove account, Mail does not delete the smtp info. You might need to do some clean up in the list to remove the extras.)
Under Advanced Tab:
Apple defaults to the default ports (25,465, 587) but some servers prefer a custom port. Check to see what port they recommend. Gmail for example likes 465, Outlook.com 587.
Verify if your server require SSL. Both Gmail and Outlook.com require SSL.
User Name. Enter full email address eg. UserName@gmail.com
Add password if it is blank. (If you have two step authentication setup, you'll need to get an app specific password more info)
Open Connection Doctor under Window in the Menu bar.
Do you see green light now for your account SMTP?
Gmail seems to work best in Mavericks iyou delete and start clean. This involves reviewing your settings in your browser and in Mail. See
Use Gmail with Mavericks Mail **
http://www.needhelp4mac.com/2013/12/use-gmail-with-mavericks-mail/
**(Fair disclosure: NeedHelp4Mac is my site. I may receive some form of compensation, financial or otherwise, from my recommendation or link.)
