OK I don’t know much about delphi but I have done a few projects and normally have typical windows component styling like so:

I was working on a project that I started from some demo code, however, and the style is completely off, as can be seen from this button:

I am not sure why it is different as I am using identical components. Is there some project setting I can change to get my normal stuff back on this application?
Yes. Open
Project -> Options -> Applicationand check the “Enable runtime themes” box and that should fix it for you.