Your web mail has to provide either POP3 or IMAP in order to use a mail client. I don't know if Yahoo allows this. In GMail, you need to login in to your account and go into the settings to allow either POP3 (most common) or IMAP. Once this is done, in thunderbird you can create a new account and there is a setting to import/use your gmail account. You'll need to create an individual email account in thunderbird for each of the locations you want to pull mail for. You will need information such as user name (email address), password, and port number (if it isn't a default number) in order to create some of the thunderbird accounts for those mail boxes.
Only POP3 mail is stored (downloaded) on your drive. IMAP leaves all mail on the server so that it can be accessed from any computer you have a mail client setup on. The mail server protocol dictates which protocol to use on the client. Gmail supports both so you can choose which one you want to use. Other services may only use one protocol.
Everything else works as expected - like the web based mail. Messages go to you inbox. You can create new folders/boxes and move messages to those folders/boxes.