If you've ever started a Gmail, Hotmail, or other email account, chances are you've used webmail. If you work in an office and use a program like Microsoft Outlook, Windows Live Mail, or Mozilla Thunderbird to manage your emails, you're using an email client.
Both webmail and email clients are applications for sending and receiving email, and they use similar methods for doing this. Webmail is an application that is written to be operated over the internet through a browser, usually with no downloaded applications or additional software necessary. All of the work, so to speak, is done by remote computers (i.e. servers and machines you connect to through the Internet).
Email clients are programs that are installed on local machines (i.e. your computer, or the computers in your office) to interact with remote email servers to download and send email to whomever you might care to. Some the back end work of sending email and all of the front end work of creating a user interface (what you look at to receive your email) is done on your computer with the installed application, rather than by your browser with instructions from the remote server. However, many webmail providers allow users to use email clients with their service.