You can read your online e-mail using NeoMail, Horde, or SquirrelMail. All of these popular web mail applications allow you to read your e-mail, save them, keep an address book, and perform all of the other basic e-mail functions that you are used to. The main difference between web mail and an offline e-mail applications, such as Eudora or Microsoft Outlook Express, is that all of these functions are performed online, rather than on your own computer, and offline applications generally provide more features.
Web mail is optional. You can still use your favorite offline e-mail applications, if you prefer. You can even use a combination of web mail applications and another tool - you can check your e-mail online, perhaps when you're at work, and download your e-mail to your computer when you get home. You can automatically configure Outlook Express, or manually configure other mail applications to read your e-mail offline.