Thunderbird is an e-mail client ... it interacts with the account server to access the mail content. It supports both POP3 (Post Office Protocol 3) to download messages OFF the server and store them locally on the user's computer, and IMAP (Internet Mail Access Protocol) to view the content on the server and sync the server-side folders with the client-side folders. Each mailbox account set up in it can be separately configured for the desired protocol.
IMAP is preferable when an e-mail address is accessed from multiple devices, but it requires more housekeeping.
POP3 is preferable when only one computer is used. Free mail services (Yahoo, HotMail, GMail, etc.) usually don't allow POP3 unless a subscription is paid for premium-level access. That doesn't make sense to me since much more server-side storage space is required when mail is not downloaded off, but I suppose the reason is that forcing the user to access via web or IMAP allows piggybacking advertising content onto the user's mail which provides a revenue stream to the host.
E-mail tends to be the Internet function that people have the most difficulty understanding because there are so many different ways to utilize it and related parameters and functions. I deal with customers often regarding these issues. POP3, IMAP, incoming servers, SMTP, outgoing servers, mail clients, web clients, server-side folders, client-side folders, downloading messages off the server, keeping messages on the server, checking mail from computers, checking mail from smartphones and PDAs and tablets, mailbox forwarding, spam detection, white lists, black lists, SMTP Authentication, SSL and encryption, port numbers, etc.