In most cases, your email account exists on a central server, and not on your personal computer or workstation. An email client is software that allows you to read and send email through your account by connecting to this server. Email clients generally provide a simple interface through which you can access your email account.
More detail. It's called a client because e-mail systems are based on a client-server architecture. Mail is sent from many clients to a central server, which re-routes the mail to its intended destination.
Many email clients are available. Some in common use are Thunderbird, Mac OS X Mail, Outlook Express, Entourage, Outlook. Webmail is a special kind of email client that can be used to access email account anywhere in the world (as long as it has an internet connection).

Mozilla Thunderbird 3.0 email client

Microsoft Outlook 2007 email client

Apple Mail email client

Microsoft Outlook Express 6 email client