In simple words, Client-Server means that one part of the software acts as a server - typically non-visual application seating somewhere in a big black box that takes requests from and _serves_ results to clients (hence the term "server"). Clients are typically desktop programs talking to those server.

For example, when you get money from ATM machine at the bank, ATM machine usually runs client program that talks to bank's server.

Using Outlook (client) to get/send mail from mail server is another example.

Web-based architecture these days usually means that all or most interaction of users with the system is performed via web interface - i.e. they use their web browsers to talk to the application which seats on the web server, whereas what is going on internally on the server is usually not known to us.

Good examples of purely web-based applications are Yahoo services such as Yahoo Answers or Yahoo Mail. Sometimes such applications offer web-based services intended not only for humans, but for other programs - like for example Google Maps - they have APIs that let your programs use their services via web interface.

The difference between the two is often blurred, though. Web browing itself is an example client-server technology, where browser is a client and the web server is well, a server.