An email client is a program installed on your computer that checks your email for you, the most famous client is Microsoft Outlook.
Outlook is a full featured program and comes as part of the MS Office suite of productivity software. However, all computers come out of the box with a free email client. Windows used to call their free version Outlook Express and with Vista has changed the name to just "Mail." Mac's application is also called "Mail."
I find that email clients are unusually associated with law firms with a tech department, or with lawyers with some sort of tech savvy. My colleagues often tell me that they never "check" their email and I see reason to ever have to check anything and highly recommend using an email client and pairing it with IMAP service.
I think this is best for several reasons.
First. It saves the time and effort of launching the web browser every time we want to check the mail.
Second. Mail clients also alerts us when we have a new message as soon as it comes in, therefore there is no need to make a specific effort to check email. Because the client check email for us, we can open the important messages with a single click. (If you are using Gmail, you can download a alert feature that does the same thing without a client)
Email clients also integrate nicely with calendars, contacts, and other software on the computer. Whether its MS Office or the standard free stuff. (After using full featured Outlook and Outlook Express, we prefer Apple's Mail, Calendar, and Address Book; although Outlook is pretty good once you put the time in to get to know it.)
Third. My favorite feature of the clients are the calendars and the associated alerts. By using a client we are in constant contact with the calender and I always schedule an alerts for appointments as email, or screen pop up, either minutes, hours, or days before.
I have found the alerts invaluable for court dates and service deadlines far in the future. The client's integrated calendars are easy to use and have features for beyond simple alerts. For example, the clients will also generate and send email to other people you want to attend the event.
Of course, most all of these clients can easily be synced with a cell phone. Blackberry does this well and is the most popular choice. However, after owning one for while and being very impressed with it, we tried the iPhone and found it better in every way that matters and in many ways that don't matter all that much.
It's essentially the same relationship between Macs and PCs. PCs and Macs do essentially the same things, but getting a Mac to do something is simpler, faster, more stable and requires far less configuration than a PC. This includes things that are normally thought be the done only by experts, including publishing web sites. Mac's ease, speed, and simplicity gets more done and encourages trial that normally ends in surprising success and not error messages, waiting, and frustration.
I suspect that many lawyers don't know how to set up a client with their email service but it is not difficult. Clients come with wizards that can set up most free email service like Gmail, Hotmail and Yahoo with only minimal information. i.e. your email address and password.