If your account USED TO use POP, those emails are stored on your computer. Only ONE reader is allowed (no iPhone pickup), or 'already Read' indicators will be bonkers.
If your account NOW uses IMAP, the email bodies are ALL stored on the Server (on the Internet) and only the Headers are sent to your computer initially. When you want to read one, the body is downloaded to you to read. When you mark as 'Read', the internet copy gets tagged as 'Read' so that all Readers (Mac AND iPhone) see that email as 'Read'.
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Several of the major email Providers have increased their security levels, and are requiring more complex login sequences, and replacing "just" password with a complex token scheme. These services, including Google/gmail, AOL mail, and Yahoo are harder to set up properly, and your initial account setup MUST invoke a Web browser interface to continue the setup process and convert to token authentication. "Just password" has been discontinued.