Creating box excerpts – confused

First time here and am certainly no expert when it comes to wordpress so any advice would be greatly appreciated.

I have done a lot of research regarding post excerpts and still cannot find the information that I am after. My website is using the Responsive theme, however the excerpt function does not appear to work. After searching for a decent plug-in, none of them seem to work with the theme that I am using.

I am trying to create exactly what the author of this website has done however I do not want my post excerpts to appear on the homepage as I have a separate page for that. Do I need to use widgets, should I be using codes (if so, then where do I need to change/write the code).

What should I do, any advice would be greatly appreciated.