To enable chat:

Click the messenger icon between your Yahoo! ID and Sign Out up at the top of the page.

This opens a pull-down menu. Click Available, Busy, or Invisible to enable chat. (For a few seconds, you'll see the message Loading chat…. Then you’ll see the availability you just selected.)

Now look under Contacts on the left—click any name in the list under # Online to start a chat with that person.

Until you make a change to Available, Busy, or Invisible, Yahoo! Mail will keep what you selected… even when you leave Yahoo! Mail and come back. If you choose Sign out of chat, you’ll remain signed out until the next time you activate Yahoo! Chat.

Yahoo! Messenger users: If we detect that you’re already signed into Yahoo! Messenger on your desktop, we'll just quietly not sign you into chat; we don't want to interrupt what you're already doing. Of course, you can always change your status from the menu in Yahoo! Mail. Signing into chat will sign you out of Yahoo! Messenger on your desktop.

Useful links:

How do I sign out of or disable chat?

As a network administrator, how do I disable Yahoo! Mail's chat feature on my network?