I'm not really sure how or what you've configured, but all that seems to be required is the incoming and outgoing mail servers, your username, and password. The latter are your Verizon account online username and online password.
Open Mail and select Preferences from the Mail menu. Click on the Accounts icon in the toolbar. If you have an account you've already tried that isn't working, then delete that account and any other account you don't use - for example if there's a .Mac account listed and you don't have a .Mac account, then delete it.
This all assumes you have already set up an email account with Verizon.net and have an email account address and password. The following is from Verizon's Help page:
Setting up your Verizon Online email with Mac OS X Mail
Start OS X Mail.
The OS X Mail program opens.
From the Mail menu, select Preferences.
Click on Accounts icon in the toolbar.
The Accounts window appears.
Click Add Account.
The Account panel appears.
Note: Depending on the version of OS X that you are using, it could either be a button titled Add Account or a plus sign "+" at the lower left of the window.
Click the Account Information tab.
From the Account Type menu, select POP.
In the Description field, enter a name for this account.
In the Email Address field, enter your Verizon Online email address (format is username@verizon.net).
In the Incoming Mail Server field, enter: incoming.verizon.net
In the User Name field, enter your Verizon Online user name.
In the Password field, enter your Verizon Online password.
From the Outgoing Mail Server menu, select New.
The SMTP Server Options screen appears.
In the Outgoing Mail Server field, enter: outgoing.verizon.net
Note: Leave all other settings in the SMTP Server Options screen in their default values.
Click OK.
In the Accounts window, click the Advanced tab.
Check the Enable this Account option.
Check the Include when automatically checking for new email option.
Configure any other email preferences you would like to set for this account.
Click OK.