It is a common practice to forward or redirect emails to external email accounts like gmail.com, outlook.com, yahoo.com etc.
Consider that you have an email account named firstname.lastname@example.org and you have setup a forwarder(email redirect) from email@example.com to firstname.lastname@example.org.
Now, any email sent to email@example.com will be forwarded to firstname.lastname@example.org. In these days, the email account email@example.com will essentially receive a lot of spam emails as well. Now, these spam emails will get forwarded to your gmail.com, yahoo.com or outlook.com email account. This is dangerous as gmail.com, yahoo.com & outlook.com will consider the email account firstname.lastname@example.org & its email server as source of spam even though original source of spam is not your email account or your server. This way you server will get blocked in gmail.com, yahoo.com & outlook.com mail servers and they will refuse to receive any email from your server.
Definitely it is very convenient to see all your emails in your gmail.com or yahoo.com or outlook.com email. It is possible to get all of your emails in your gmail.com or yahoo.com or outlook.com email account with help of a feature called "mail fetcher". Usage of "mail fetcher" will prevent above mentioned danger associated with email forwards(or email redirects).
Following links will help you to configure "mail fetcher".
If you have an email account other than these 3, please contact provider of that email service to see how you can download pop emails into that account.