There are two popular mail standards in use today with various tradeoffs.
POP (sometimes called POP3) is the most standard and common protocol. With POP email your email client connects to the mail server, checks for new mail, and if there is new mail will download it and delete the message from the server. This works well when you only want to access email on one computer.
IMAP is quite well supported now and has many advantages. With IMAP your email client connects to the server and if there are new messages will download them but not delete them. This allows you to view the same inbox on multiple computers/devices/webmail. If you move something to a folder on one computer it will appear in that folder on every other device. The disadvantage of IMAP is that all messages are stored on the mail server and your inbox is limited to 1GB of storage. This limit cannot be changed and if it is exceeded messages will be returned to the sender of the message.