There are two main email protocols used to communicate with email servers, IMAP and POP (actually POP3 these days).
When you have an IMAP account, actions you take on one device, such as reading, deleting or moving a message, are performed on the server and are therefore seen by all devices connected to your email account via that same protocol. That means when a message is read on one device, it will be marked read on all your devices, but all email should appear on all devices and when a message is deleted, it should be deleted on all devices.
POP3, an older protocol, downloads all new email messages to your device you are using to check email and does not sync. This is problematic when you have multiple devices, as default behavior for most POP3 clients is that when an email message is downloaded via POP3, it is deleted from the server at that point so depending on which device you checked your email on, you could have one set of messages on your iPhone, one set on your iPad and one set on your desktop, and would have to check all your devices for a particular email message. Some mail clients do not auto-delete messages, and so you would instead see new email messages on all your devices that seemed to be duplicates; when you read and deleted the email on one device, it would still show as a new unread message on others. Apple Mail allows you to specify either behavior.
You may want to look at this page and try to narrow down what your issue(s) may be:
Mail - Official Apple Support