An HTC file is a web file format file used by Microsoft Internet Explorer. It contains definitions for HTML components that perform small functions for performing different tasks inside the web browser. Functionality defined inside HTC files is reusable and is accessible across multiple HTML pages.
HTC File Format - More Information
HTC files are saved in plain text file format. This means you can open these in any text editor and make changes to the functionality. These are usually opened in Internet Explorer, but that way you will only be able to view these files and will not be able to edit them.
An HTC file is actually an HTML file that is a combination of script and a set of HTC-specific elements for defining the components.