Because you not yet configured the “Leave a copy of message on server” option in your POP email client, so email is deleted after POP email client retrieval. If you need to leave email on server for viewing by Web interface (or by Mobile phone via ActiveSync), it is important to set leave a copy in server, or it is more recommended to use IMAP protocol instead of POP.
It is recommended to configure your PC email client to use IMAP synchronization protocol and the iOS/Android mobile devices to use ActiveSync protocol (select “Exchange” or “Outlook” account), they are synchronization protocol so your action on the email is reflected immediately in Web interface and other devices (event like new arrival/read/delete/sent, etc.)
IMAP protocol will synchronize the folders between PC and server, so if your PC email client default folder name mismatch with that of the server folder name, the new name will be created at each side. Normally PC IMAP email client allows you re-map the folder name to that on server, customer has to adjust this on his/her PC IMAP email client. (e.g. Outlook Express and Windows Live Mail)