They must be talking about the form validation. Client side validation is usually done with JavaScript/jQuery which can be easily manipulated using tools like Firebug. If user disables his JavaScript, the validation will fail and wrong data will be posted, inorder to prevent that, Server Side Validation is used for example, PHP, ASP.
The data is posted to the server, server validates the data, if it's incorrect, it will throw back the error to the user, and the most important is that server side validation cannot be manipulated/modified easily.
Now you may ask why to validate Client Side if it can be modified/broken easily? Well, not everyone is intelligent to fool your validation, home/normal users often forget some fields to fill up which are compulsory, if you do not keep a client side check at all, your server will get more and more requests directly, thus increasing the load, inorder to prevent that, you need to keep a client side check as well, which will prevent the form details to be posted to the server directly, thus decreasing the server load.
Also, it will save the time for your users, JavaScript validations are very quick, the users are responded with the related error messages when they are typing, or they move to the next field or very soon after the submit button is pressed. Where as server validation will make the process lengthy as forms will be submitted first, server will validate, and than it will return the message. Not much friendly for an user huh?
So the bottom line is validate both, Client Side and Server Side, but make sure you DO VALIDATE Server Side.