Here’s how:

(This is a slightly long-ish procedure, but it’s not difficult. So take a deep breath…)

1. Click Options in the upper-right corner of the page and select Mail Options.
2. From the list on the left, click Accounts.
3. In the middle area of the page, click Add or edit an account.
4. The “Mail Accounts” page opens. Click Add.
5. In the box, enter an account name for your own reference (for example, Work, School, The One Where My Name Is Lenny, etc. – no one else will see this) …and click Continue.
6. Look now: two boxes! In the first one, put the Name you’d like others to see when they get email from this address. In the second one (this is the part you’ve been waiting for), type the Email address you’d like send from – from within Yahoo! Mail! (You’ll be asked to verify this other address in an email we will send to you there.) Click Continue.
7. Do you want to check email in this account from your Yahoo! account as well? (You can check your mail in clients like Outlook and Eudora, but currently you won’t be able to check mail in other webmail accounts with providers like Hotmail, Gmail or Yahoo. You CAN SEND from these other addresses from within Yahoo! Mail, however!)
1. If YES (you want to check mail in your Outlook or Eudora or similar client), skip to step 10. Yep. Go ahead. Step 10. Down there ▼.
2. If NO – you don't want to check this account, just SEND from it from inside your Yahoo! account – click Skip this Step at the bottom of the "Add Account" page and continue to step 8.
8. On the page that opens, check to make sure you’ve entered the correct email address. If you need to make any changes, click Edit Address. If it looks fine, click Send Verification to generate the email we send to you to make sure this is an actual account you use. We gotta do this to help prevent spam!
9. The “Verify Account” page opens. Here’s the easiest way to complete the setup and start sending from this other address:
1. Leave this Verify Account page open.
2. In a different browser window or application, sign into your other Mail account.
3. Look for an email from Yahoo! Member Services, and open it.
4. Copy the “confirmation code” in the last line of the email by either double-clicking it or clicking and dragging over it, then either clicking Copy from the Edit menu or pressing Ctrl + “c”. (Mac users press Command + “c”.)
5. Get back on the “Verify Account” page and Paste that confirmation code in the box beside “Confirmation Code:” …You can paste by clicking Paste in the Edit menu, or by pressing Ctrl + “v”. (Mac users press Command + “v”.)
6. At the bottom of the page, click Verify Account.
7. OK, last step! A sign-in page opens with your ID already filled in. Sign in with your password.
8. Hot, you’re done. See the new account in the list on your Mail Accounts page? Now you can easily send from it by opening a Compose page, clicking the down-arrow on the end of the “From:” box, and clicking the new account in the pull-down list of email addresses. That’s it! People receiving the message will see it as being sent from that address.
10. OK, so you also want to be able to check email in this other account from your Yahoo! Mail account? No problem. On the page you’re looking at, enter that account’s mail server name (currently, only POP mail servers are supported). It will be something like pop.goestheweasel.com. To find out what it is, either search that application’s Help pages for “POP” or “server” (or both), or contact that email provider directly.
11. Next, type your user name for that account and the password. This is completely confidential! Nobody has access to this information, but Yahoo! Mail needs it to check the account. Note: Your user name is the text in your email address before the "@" symbol.
12. We thought a good way to identify mail from different accounts would be to give it a different color in your Yahoo! Inbox. So that’s what this last part is about: pick an indicator color to mark messages received from this account.
13. Click Setup Mail Server. Excellent! We’re done with this part. You should now be looking at the “Verify Account” page. Go back to Step 8 to hopskotch your way through account verification. Once that’s done, you can do it all from Yahoo! Mail – check mail in that account and send mail from that account. Enjoy!

Note: Yahoo! Mail only retrieves messages from your external (POP) mail accounts when you request it. Do this by clicking the down-arrow on the Check Mail button in the upper-left of the page, and then clicking either Check all accounts, or clicking the name of the other account’s server.