The more commonly used fields in Webform (text, radios, etc.) offer the ability to set a custom Required message string. This is the text that shows in the error message when a required field is not filled.

Here’s what that screen looks like when you’re configuring a field on the Build screen:

There are several element types that do not provide a UI field to set this Required message value. The composite Name and Advanced Address field are two examples I encountered. For these fields you have to dig a little deeper, but it’s easy once you get the hang of it. Visit the Advanced tab for the field, then scroll to the bottom where you’ll see a Custom Settings fieldset.