The way your email client is set up on your phone or tablet can determine how email is handled on your desktop as well. For instance, you may find that after downloading all of your email to your phone, the email disappears on your desktop. This behavior can be changed by either using IMAP (Internet Message Access Protocol) or POP3 (Post Office Protocol version 3).
When email is set up using POP3, or "Post Office Protocol," the email will be deleted from the server when it is downloaded to the client (or your email app on your phone or tablet). The USPS is a good analogy for POP3 -- once you pick up your mail from the post office, or once it's delivered to your mailbox, the post office doesn't have it anymore. They don't keep a copy of all of your physical mail, and it works the same way with POP3 email.
IMAP, or "Internet Message Access Protocol," keeps your email on the server. It allows multiple clients (or email apps) to be simultaneously connected to the same mailbox. This means the email is not downloaded to your device, and any change you make to a message in your email client will be reflected on the server and vice versa. IMAP is a more modern method of retrieving email. You can choose between IMAP or POP3 in your email provider's settings.
If you're using an internet service provider's email service and you're having difficulty with it, you can easily switch to a web-based service like Gmail. In fact, Gmail will actually retrieve the email for you from your current provider. Find out how to set this up here (support.google.com).