Troubleshooting is generally made easier when you understand how something is supposed to function in the first place. Apple's Mail application is no different. To successfully send and receive messages, Mail must have four major functions working: the message, the application, the Internet connection, the e-mail server. Resolving issues with the Mail application generally falls into one of these areas.
The Message. In order for Mail to send your message, you must have a correct e-mail address entered for your recipient. Attachments must be smaller than the allowable size (based on restrictions provided by your mail service provider as well as the recipient's).
The Application. The Mail application must have the correct settings for your e-mail account(s). This includes your incoming and outgoing server addresses and your username and password.
The Internet Connection. Your Internet must be functioning and configured properly for email to be sent.
The E-mail Server. Your e-mail service provider must be functioning properly and able to access and accept your username and password.
Update your software. Updates are constantly becoming available for Mac OS X. These updates are designed to resolve issues with applications like Mail. Use Software Update to check for any new updates and apply them. If Software Update is able to check for updates, you know that your Internet connection is not the issue with Mail. Once you have applied all the updates, check the issues with Mail. If you continue to have them, try downloading the latest Mac OS X Combo Update package from the Apple Downloads page.
Check your connection. Mail has a built-in troubleshooting tool called Connection Doctor. From the Mail menu bar select Window > Connection Doctor. The window that pops up will show you the status of your mail account(s). Green dots next to each connection point indicate everything is working fine. Red dots indicate there is an issue with the connection. This will help you determine if the problem lies with your Internet connection, your e-mail server, or the application itself.
Check the message. Often, the problem with an e-mail not being able to be sent is within the message itself. This can occur if the recipient's email address is not valid. It is possible that their e-mail address may have been entered incorrectly, but autofill retains it as a previous recipient. Check your previous recipients list by right-clicking (Ctrl+click) on the recipient's address. If the address is incorrect, choose the option to "Remove from Previous Recipients List." You can also access the entire list from the Mail menu bar > Window > Previous Recipients.
Confirm your ESP (e-mail service provider) settings. Your ESP (Gmail, Yahoo!Mail, Hotmail) can be different than your ISP (Comcast, Cox, Qwest), though many ISPs also offer e-mail services. In any case, be sure your settings are correct by opening the Mail Preferences (Command + ,) and clicking on Accounts. If you have recently changed your password online, you must change it in your Mail account settings as well. If you are still having issues, Apple has a great cheat sheet to help.
These are just a few of the possible issues that could result in a message not being sent or messages not being received. If you have a specific issue not covered in this article, Apple's KB article, or any other support articles here on MacFixIt, use our Contact page in the right column and let us know.