How do I get the same emails on both iPhone and computer?

You need an IMAP account to do that, sounds like you have a POP account. POP is designed to access mail on one device. To get things in sync on multiple devices requires an IMAP account. So, see if your email provider offers IMAP, if not get a gmail account(can be setup as IMAP) and forward your emails to your gmail account.

I used to have a NTLWorld account(now taken over by Virgin and at the time they automatically deleted any e-mails older than 30 days.
It's worth checking that is no longer the case as some of the accounts have been migrated to the GoogleMail servers so it shouldn't be an issue now.
Better to be safe than sorry though

As already stated, what you are observing is a characteristic of POP3 mail servers. With POP once an email is retrieved from one client it is normally deleted from the server.

With IMAP the master copy of the email is always on the server, and the clients (iPhone or computer) merely replicate what is on the server. If you delete a mail from one device it is deleted from the server, and then from other devices when they next sync.

So if email disappears after being retrieved by a device and you have not deleted it from that device the client is not configured as IMAP, whether or not the ISP supports IMAP.

In IMAP, actual emails are not downloaded to the various devices, what you see are merely pointers to the actual emails which remain on the server. Your email provider may support IMAP, but it appears you have not configured email on your devices as IMAP. It appears your mail is configured as POP.

The above post is somewhat misleading.

Imap email is indeed downloaded to the device.

Imap email are simply stored on a server and not deleted until you delete them, not withstanding your service providers policy regarding retention.

Each device "syncs" with the server and keeps it's own copies of email on the device.

Think of it like this...your first email may be "email 1". Each device that is configured will check the server and say "hey, I don't have email 1, I better download a copy".

Once you read email 1 on any device that device will mark email 1 as read and next time the other devices check they will all mark it as read.

Once you delete it on any device the other devices will see it has been deleted and also delete their copy.

The above post implies that you always need to be connected to your imap server to read your emai, not the case.

You can read your local copies offline.

If you do not have access to imap, you can set it up as follows.

Set your home computer email to leave a copy of your emails on the server, sometimes stated as "do not delete mail from server".

You can usually set it to delete after a certain number of days.

Set your other devices to "never delete from server"

This gives all your devices time to download messages before they are deleted.

Unlike imap, this method does not mark messages on server as read, forwarded, replied to etc. So email will only be marked as such on the device the action was taken on.

Imap email is indeed downloaded to the device.

Not true. E-mail clients using IMAP generally leave messages on the server until the user explicitly deletes them. Clients may store local copies of the messages, but these are considered to be a temporary cache.

You are just playing with words and quoting Wikipedia.

I agree totally that messages are left on the server...I said that above.

You claimed that emails are "pointed to" implying they are not on the device.

How can you say emails are saved on the device as a temporary cache and also say they are not downloaded to the device???

Fact is, I can turn off my internet connection at home, on my ipad, on my iphone, on my laptop, and read any email that I have received through my IMAP server even if it has not been marked as read before.

You can read your local copies offline.

This depends on the settings for the email client used to access an IMAP account. With the Mail.app on a Mac for an IMAP account, there is a setting under the Advanced tab for account preferences for "Keep copies of messages for offline viewing" with the following options.

All messages and their attachments.
All messages but omit attachments.
Only messages I've read.
Don't keep copies of any messages.