Sounds like you are using a POP3 account. This type of account will deliver the email to the first email client that fetches them, and unless you set your options to not delete them from the server, will delete them thus making them unavailable for other devices to fetch the same email. The setting to not remove them from the server on your iPhone is usually under Mail,Contacts,Calendars>your account>Advanced.
If you switched to an IMAP account such as Gmail, Yahoo or iCloud email this would not happen.
I have a gmail, yahoo, icloud and my main account with wildblue (which is a home satalite service) and all of these accounts are linked together on the phone with wildblue, yes wildblue is a pop account. Not sure on changing the home computer settings...I will try the gmail account.