Mac OS X 10.0 Cheetah
By default, Mail.app tries to synchronize each IMAP mailbox you define for your mail accounts each time you open the application. This includes the download of every e-mail with all attachments in each of the mailboxes. Both synchronizing and downloading will take time, in particular over a slow connection.
For large mailboxes (more than a few hundred e-mails or more than a few hundred megabytes, whatever applies), this behavior of Mail.app doesn’t scale well. To speed up Mail.app, the automatic download of e-mails with their attachments can be disabled. Since IMAP accounts have their data stored on the mail server, you don’t run into danger of loosing e-mails (as long as the IMAP server doesn’t loose them).
To disable the automatic download of e-mails, open the Preferences… dialog of Mail.app and select the Accounts pane. Within that pane, select the IMAP account holding the problemantic mailboxes and select the “advanced” subpane. In the popup menu in the middle of the subpane, select the item “Don’t keep copies of any messages”, like here:
When leaving the dialog, make sure to confirm the change you just made.