Open System Preferences > Internet Accounts then select AOL on the right.

Type in the your name, email address you use with AOL and your AOL password then click Sign In.

Hopefully you won't be prompted to do that again.

I tried the solution below several times without result. 

After putting up with the issue for days awaiting my IT director to solve this, I finally deleted the AOL mail account.  Then added it back again. 

For me this fixed the problem immediately across all my apple devices. 

Use Mail Connection Doctor

Mail Connection Doctor can help explain why the Mail app isn't sending or receiving email on your Mac.

If you can’t send or receive email on your Mac, you can use Mail Connection Doctor to learn more about the issue. Mail Connection Doctor checks the status of your Internet connection and each of your email accounts.

Open the Mail app, then choose Window > Connection Doctor.

Check Internet connection status

The top of the Mail Connection Doctor window shows the status of your Internet connection. If you see a red dot, Mail can't connect to the Internet. If Mail can’t connect to the Internet, it also can't use any of your email accounts to send or receive email. Click the Network Diagnostics button to start a detailed test of your Internet connection, then learn about solutions for connecting to the Internet.

Check email account status

If you're connected to the Internet but Mail Connection Doctor shows a red dot next to one or more of your accounts, check the Details column for messages such as the following:

“Trying to log in to this . . . account failed.”

Your email service may be experiencing a service outage. Contact them or use their status webpage to make sure that their mail service is online. You may also get this message if the password for your account is missing or no longer valid.

“Could not connect to this . . . server.”

The settings for your email account may not be correct. Check Mail preferences to make sure that your email account settings are up to date. You can use the Mail Settings Lookup tool to check the settings for many email providers, or get them from your email provider.

You may also see this message if a network firewall is blocking the connection to your email service. That can happen if you installed third-party firewall software, or you're on a different network that doesn't allow the necessary email connections. If you're connecting from your corporate network, contact your corporate IT department for help. If you're connecting from somewhere else, such as a coffee shop or Internet cafe, check with their staff.

Show more detail

Mail Connection Doctor can show even more detail about the connection:

• Click the Show Detail button to open a drawer showing basic communication between the Mail app and your email service. If the drawer is empty, click Check Again to start another check. You can copy data from this drawer and paste it into another app for easier viewing. Messages that contain "ALERT" come directly from your email provider.
• Select the Log Connection Activity checkbox to cause Mail to generate detailed logs of its communication with your email service. To view the logs, click Show Logs. To stop logging, deselect Log Connection Activity.

Last Modified: Apr 7, 2015

30% of people found this helpful.

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This solution works.

After upgrading from Yosemite to El Capitan, all the previously imported email accounts work (Gmail, iCloud, Yahoo, Exchange, Hotmail). AOL was the only one that continued to request a password. I also verified online that I could login using the web interface. This in itself verifies your internet is working, the username and password are correct and that the AOL service is not down. Being able to retrieve other accounts through mail also verifies Mail can connect to the internet. That leaves Mail with probably having not imported AOL connection parameters properly during the upgrade process. Deleting and re-adding the account then causes Mail to build a new preference file based on fresh connection settings.

Do not bother with the other canned response. It generally only assumes everyone is an idiot that doesn't know how to use their computer.

Same experience.  The mail connection doctor, mentioned in the post after yours, was worthless, as it only suggested obvious things that I'd tried already.  But deleting the account and adding again from scratch did the trick.

I have had the same issue and tried the connection doctor, tried by removing AOL on the System Preferences page and then re-installing it again, this worked and AOL now accepted the same password it was rejecting before so happy days or so i thought 1 week has passed and today AOL is again asking me for my password and again it is saying unable to Verify username or password ! WTH ! I have just removed AOL for the 2nd time and it has again worked ! It has to be something within the El Capitan as WEB AOL is ok as is my AOL on my android Sony Z3.... Please sort this APPLE !

Dont know if anyone else has noticed how the updated OS has slowed there MAC down? I have had to upgrade my RAM just for it to respond like it did prior to the update.

Has any fix occurred for this?  I have multiple email accounts and all but aol works fine.  I have deleted the account and reestablished it and it only fix the problem for a short time.  Connection is good.  It just keeps asking for aol password over and over. 

I have similar issue but with hotmail. I have chosen to delete the account and recreate again, which didn't stop the annoying problem from coming back.

In my case, I found El Capitan handles it better than previous version. The asking for password is just temporary. I chose to ignore, and I can actually get email from that account.

But the message drives me crazy. It pops up many time a day. Apple should fix this.

Yes, I have this same issue with hotmail.  It just started.    I occasionally have the same problem with Yahoo, too, but that's been going on for a while.

Thank you, thank you, thank you!  This happens repeatedly to me.  Sometimes I ignore it, and it works anyway, but today I couldn't send any mail.  At least now I know what to do.

I have comcast.  I send and receive emails even as Apple tells me my password is not correct.  This is so annoying.

I have tired several of the solutions, deleting the account, etc., nothing seems to work.

appears to me that someone in Apple should be hearing about this and making a fix.

Tim

i started having this problem this AM with Comcast e-mail....so far no solution.....it seemed to happen when a Notification of mail came in...then immediately computer began asking for my password and even though i supplied it... the computer responded that it was unable to verify.

I might add that Connection Doctor is/was useless......but I have noticed there may be a connection of some sort to my recent  iTunes store activity. I also deleted all social activity notifications which i do not use. I am thinking the problem is something in the Notifications permissions, an app i do not need and did just as well without in the past.