You can read your online email using NeoMail or Horde. Both of these popular webmail scripts allows you to read your email, save them, keep an address book, and perform all of the other basic email functions that you are used to. The main difference between NeoMail and an offline email 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 many more features.
NeoMail or Horde is optional. You can still use your favorite offline email applications, if you prefer. You can even use a combination of NeoMail or Horde and another tool - you can check your e-mail online, perhaps when you're at work, and download your email to your computer when you get home.