You need an IMAP account.
With an IMAP account, all server stored mailboxes are kept synchronized with the server automatically with each email client used to access the account.
For received messages with a POP account, there is option to not remove the messages from the server when downloaded by each email client used to access the account. With the Mail.app on a Mac, this is available under the Advanced tab for the account preferences. There is a Never option for POP account preferences with the iPhone's Mail client.
You can email this message with an attachment to yourself to be received by the Mail.app on your Mac after selecting Never for removing messages from the server when downloaded by the iPhone's Mail client.
On your iPhone, go to Settings > Mail, Contacts, Calendars.
Select the POP account.
Select Advanced for the account settings.
Under the Incoming Settings section, select Never for delete from server.